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4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def /Mbbox { 0 0 moveto false charpath flattenpath pathbbox newpath } bind def /Mmin { 2 copy gt { exch } if pop } bind def /Mmax { 2 copy lt { exch } if pop } bind def /Mrotshowa { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix Mshowa /Mtmatrix exch def /Mrot 0 def } bind def /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def 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sub mul sub -2 div } bind def /MathScale { Mwidth Mheight Mlp translate scale /yscale exch def /ybias exch def /xscale exch def /xbias exch def /Momatrix xscale yscale matrix scale xbias ybias matrix translate matrix concatmatrix def /Mgmatrix matrix currentmatrix def } bind def /Mlp { 3 copy Mlpfirst { Mnodistort { Mmin dup } if 4 index 2 index 2 index Mlprun 11 index 11 -1 roll 10 -4 roll Mlp1 8 index 9 -5 roll Mlp1 4 -1 roll and { exit } if 3 -1 roll pop pop } loop exch 3 1 roll 7 -3 roll pop pop pop } bind def /Mlpfirst { 3 -1 roll dup length 2 copy -2 add get aload pop pop pop 4 -2 roll -1 add get aload pop pop pop 6 -1 roll 3 -1 roll 5 -1 roll sub div 4 1 roll exch sub div } bind def /Mlprun { 2 copy 4 index 0 get dup 4 1 roll Mlprun1 3 copy 8 -2 roll 9 -1 roll { 3 copy Mlprun1 3 copy 11 -3 roll /gt Mlpminmax 8 3 roll 11 -3 roll /lt Mlpminmax 8 3 roll } forall pop pop pop pop 3 1 roll pop pop aload pop 5 -1 roll aload pop exch 6 -1 roll Mlprun2 8 2 roll 4 -1 roll Mlprun2 6 2 roll 3 -1 roll Mlprun2 4 2 roll exch Mlprun2 6 2 roll } bind def /Mlprun1 { aload pop exch 6 -1 roll 5 -1 roll mul add 4 -2 roll mul 3 -1 roll add } bind def /Mlprun2 { 2 copy add 2 div 3 1 roll exch sub } bind def /Mlpminmax { cvx 2 index 6 index 2 index exec { 7 -3 roll 4 -1 roll } if 1 index 5 index 3 -1 roll exec { 4 1 roll pop 5 -1 roll aload pop pop 4 -1 roll aload pop [ 8 -2 roll pop 5 -2 roll pop 6 -2 roll pop 5 -1 roll ] 4 1 roll pop } { pop pop pop } ifelse } bind def /Mlp1 { 5 index 3 index sub 5 index 2 index mul 1 index le 1 index 0 le or dup not { 1 index 3 index div .99999 mul 8 -1 roll pop 7 1 roll } if 8 -1 roll 2 div 7 -2 roll pop sub 5 index 6 -3 roll pop pop mul sub exch } bind def /intop 0 def /inrht 0 def /inflag 0 def /outflag 0 def /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def /Minner { outflag 1 eq { /outflag 0 def /intop 0 def /inrht 0 def } if 5 index gsave Mtmatrix setmatrix Mvboxa pop grestore 3 -1 roll pop dup intop gt { /intop exch def } { pop } ifelse dup inrht gt { /inrht exch def } { pop } ifelse pop /inflag 1 def } bind def /Mouter { /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def inflag 1 eq { dup 0 lt { dup intop mul neg /yadtop exch def } if dup 0 gt { dup intop mul /yadbot exch def } if pop dup 0 lt { dup inrht mul neg /xadrht exch def } if dup 0 gt { dup inrht mul /xadlft exch def } if pop /outflag 1 def } { pop pop} ifelse /inflag 0 def /inrht 0 def /intop 0 def } bind def /Mboxout { outflag 1 eq { 4 -1 roll xadlft leadjust add sub 4 1 roll 3 -1 roll yadbot leadjust add sub 3 1 roll exch xadrht leadjust add add exch yadtop leadjust add add /outflag 0 def /xadlft 0 def /yadbot 0 def /xadrht 0 def /yadtop 0 def } if } bind def /leadjust { (m) stringwidth pop .5 mul } bind def /Mrotcheck { dup 90 eq { yadbot /yadbot xadrht def /xadrht yadtop def /yadtop xadlft def /xadlft exch def } if dup cos 1 index sin Checkaux dup cos 1 index sin neg exch Checkaux 3 1 roll pop pop } bind def /Checkaux { 4 index exch 4 index mul 3 1 roll mul add 4 1 roll } bind def /Mboxrot { Mrot 90 eq { brotaux 4 2 roll } if Mrot 180 eq { 4 2 roll brotaux 4 2 roll brotaux } if Mrot 270 eq { 4 2 roll brotaux } if } bind def /brotaux { neg exch neg } bind def /Mabsproc { 0 matrix defaultmatrix dtransform idtransform dup mul exch dup mul add sqrt } bind def /Mabswid { Mabsproc setlinewidth } bind def /Mabsdash { exch [ exch { Mabsproc } forall ] exch setdash } bind def /MBeginOrig { Momatrix concat} bind def /MEndOrig { Mgmatrix setmatrix} bind def /sampledsound where { pop} { /sampledsound { exch pop exch 5 1 roll mul 4 idiv mul 2 idiv exch pop exch /Mtempproc exch def { Mtempproc pop } repeat } bind def } ifelse /g { setgray} bind def /k { setcmykcolor} bind def /m { moveto} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /P { grestore} bind def /s { stroke} bind def /MFill { 0 0 moveto Mwidth 0 lineto Mwidth Mheight lineto 0 Mheight lineto fill } bind def /MPlotRegion { 3 index Mwidth mul 2 index Mheight mul translate exch sub Mheight mul /Mheight exch def exch sub Mwidth mul /Mwidth exch def } bind def /Mcharproc { currentfile (x) readhexstring pop 0 get exch div } bind def /Mshadeproc { dup 3 1 roll { dup Mcharproc 3 1 roll } repeat 1 eq { setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /Mrectproc { 3 index 2 index moveto 2 index 3 -1 roll lineto dup 3 1 roll lineto lineto fill } bind def /Mcolorimage { 7 1 roll pop pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index 9 index Mshadeproc Mrectproc } for pop } for pop pop pop pop } bind def /Mimage { pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index Mcharproc setgray Mrectproc } for pop } for pop pop pop } bind def /Mnodistort true def /colorimage {Mcolorimage} 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p .002 w .60938 .34125 m .59641 .3312 L .61466 .32955 L p .53 .568 .838 r F P s P p .002 w .61466 .32955 m .63135 .33934 L .60938 .34125 L p .53 .568 .838 r F P s P p .002 w .51917 .31294 m .51544 .32273 L .49258 .32213 L p .74 .7 .813 r F P s P p .002 w .51544 .32273 m .51917 .31294 L .53754 .31299 L p .703 .664 .809 r F P s P p .002 w .55506 .24369 m .54696 .24992 L .55698 .2493 L p .526 .209 .372 r F P s P p .002 w .55698 .2493 m .57006 .24276 L .55506 .24369 L p .526 .209 .372 r F P s P p .002 w .22699 .43474 m .24611 .42476 L .2714 .42858 L p .87 .948 .862 r F P s P p .002 w .28844 .41858 m .2714 .42858 L .24611 .42476 L p .87 .948 .862 r F P s P p .002 w .22033 .09696 m .23876 .10735 L .23387 .11911 L p .157 .407 .851 r F P s P p .002 w .25216 .12853 m .23387 .11911 L .23876 .10735 L p .157 .408 .851 r F P s P p .002 w .39969 .20374 m .41828 .21298 L .42687 .21639 L p .595 .07 0 r F P s P p .002 w .44394 .22488 m .42687 .21639 L .41828 .21298 L p .584 .057 0 r F P s P p .002 w .5115 .2498 m .50211 .24351 L .48798 .24248 L p .769 .393 .344 r F P s P p .002 w .50211 .24351 m .5115 .2498 L .52295 .25018 L p .726 .399 .426 r F P s P p .002 w .71415 .38931 m .69768 .37927 L .71775 .37606 L p .352 .494 .862 r F P s P p .002 w .71775 .37606 m .7365 .38594 L .71415 .38931 L p .352 .494 .862 r F P s P p .002 w .73325 .16793 m .71505 .17716 L .7152 .17071 L p .825 1 .819 r F P s P p .002 w .7152 .17071 m .73372 .16067 L .73325 .16793 L p .825 1 .819 r F P s P p .002 w .61402 .22549 m .5975 .23366 L .60489 .23153 L p 0 0 0 r F P s P p .002 w .60489 .23153 m .62336 .22278 L .61402 .22549 L p 0 0 0 r F P s P p .002 w .23876 .10735 m .25716 .11773 L .25216 .12853 L p .157 .408 .851 r F P s P p .002 w .27048 .13795 m .25216 .12853 L .25716 .11773 L p .158 .408 .851 r F P s P p .002 w .24611 .42476 m .2652 .41481 L .28844 .41858 L p .87 .948 .862 r F P s P p .002 w .30551 .40857 m .28844 .41858 L .2652 .41481 L p .87 .948 .862 r F P s P p .002 w .40864 .34867 m .42595 .33883 L .44561 .34084 L p .828 .822 .843 r F P s P p .002 w .45959 .33085 m .44561 .34084 L .42595 .33883 L p .827 .823 .845 r F P s P p .002 w .5633 .32232 m .55519 .31262 L .57026 .31186 L p .611 .608 .823 r F P s P p .002 w .57026 .31186 m .58337 .32135 L .5633 .32232 L p .611 .608 .823 r F P s P p .002 w .69768 .37927 m .68116 .36924 L .699 .36622 L p .352 .495 .862 r F P s P p .002 w .699 .36622 m .71775 .37606 L .69768 .37927 L p .352 .495 .862 r F P s P p .002 w .71505 .17716 m .69681 .18638 L .69669 .1807 L p .825 1 .82 r F P s P p .002 w .69669 .1807 m .7152 .17071 L .71505 .17716 L p .825 1 .82 r F P s P p .002 w .25716 .11773 m .27554 .12808 L .27048 .13795 L p .158 .408 .851 r F P s P p .002 w .28884 .14737 m .27048 .13795 L .27554 .12808 L p .158 .408 .852 r F P s P p .002 w .45959 .33085 m .47365 .32105 L .49258 .32213 L p .78 .747 .823 r F P s P p .002 w .50197 .31247 m .49258 .32213 L .47365 .32105 L p .778 .75 .828 r F P s P p .002 w .49258 .32213 m .50197 .31247 L .51917 .31294 L p .74 .7 .813 r F P s P p .002 w .2652 .41481 m .28428 .40487 L .30551 .40857 L p .87 .948 .862 r F P s P p .002 w .32261 .39856 m .30551 .40857 L .28428 .40487 L p .869 .948 .862 r F P s P p .002 w .59641 .3312 m .58337 .32135 L .59793 .31998 L p .53 .571 .841 r F P s P p .002 w .59793 .31998 m .61466 .32955 L .59641 .3312 L p .53 .571 .841 r F P s P p .002 w .57006 .24276 m .55698 .2493 L .56419 .24844 L p .33 0 .189 r F P s P p .002 w .56419 .24844 m .58089 .24144 L .57006 .24276 L p .33 0 .189 r F P s P p .002 w .41828 .21298 m .4369 .2221 L .44394 .22488 L p .584 .057 0 r F P s P p .002 w .46109 .23317 m .44394 .22488 L .4369 .2221 L p .565 .036 0 r F P s P p .002 w .27554 .12808 m .2939 .1384 L .28884 .14737 L p .158 .408 .852 r F P s P p .002 w .30723 .1568 m .28884 .14737 L .2939 .1384 L p .159 .409 .852 r F P s P p .002 w .68116 .36924 m .66461 .35922 L .68027 .35642 L p .352 .496 .863 r F P s P p .002 w .68027 .35642 m .699 .36622 L .68116 .36924 L p .352 .496 .863 r F P s P p .002 w .28428 .40487 m .30334 .39497 L .32261 .39856 L p .869 .948 .862 r F P s P p .002 w .33975 .38855 m .32261 .39856 L .30334 .39497 L p .869 .948 .863 r F P s P p .002 w .69681 .18638 m .67851 .19557 L .67819 .19064 L p .824 1 .822 r F P s P p .002 w .67819 .19064 m .69669 .1807 L .69681 .18638 L p .824 1 .822 r F P s P p .002 w .46109 .23317 m .47833 .24109 L .48798 .24248 L p .808 .36 .175 r F P s P p .002 w .50206 .24912 m .48798 .24248 L .47833 .24109 L p .78 .305 .106 r F P s P p .002 w .48798 .24248 m .50206 .24912 L .5115 .2498 L p .769 .393 .344 r F P s P p .002 w .53522 .30394 m .53754 .31299 L .51917 .31294 L p .701 .672 .818 r F P s P p .002 w .53754 .31299 m .53522 .30394 L .54703 .30368 L p .661 .643 .821 r F P s P p .002 w .54703 .30368 m .55519 .31262 L .53754 .31299 L p .661 .643 .821 r F P s P p .002 w .5975 .23366 m .58089 .24144 L .58636 .23988 L p 0 0 0 r F P s P p .002 w .58636 .23988 m .60489 .23153 L .5975 .23366 L p 0 0 0 r F P s P p .002 w 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.21029 L p .822 1 .826 r F P s P p .002 w .64121 .21029 m .65969 .20051 L .66018 .20473 L p .822 1 .826 r F P s P p .002 w .648 .34925 m .63135 .33934 L .6428 .33707 L p .353 .499 .865 r F P s P p .002 w .6428 .33707 m .66153 .3467 L .648 .34925 L p .353 .499 .865 r F P s P p .002 w .44329 .32911 m .46066 .31961 L .47365 .32105 L p .825 .824 .848 r F P s P p .002 w .48777 .31164 m .47365 .32105 L .46066 .31961 L p .821 .827 .854 r F P s P p .002 w .34139 .37525 m .36039 .36546 L .37413 .36856 L p .868 .948 .864 r F P s P p .002 w .39137 .35859 m .37413 .36856 L .36039 .36546 L p .868 .948 .864 r F P s P p .002 w .58089 .24144 m .56419 .24844 L .56779 .24742 L p 0 0 0 r F P s P p .002 w .56779 .24742 m .58636 .23988 L .58089 .24144 L p 0 0 0 r F P s P p .002 w .55698 .2493 m .5438 .25414 L .5474 .25371 L p .242 0 0 r F P s P p .002 w .5474 .25371 m .56419 .24844 L .55698 .2493 L p .242 0 0 r F P s P p .002 w .34885 .16919 m .36712 .17936 L .3626 .18504 L p .161 .412 .853 r F P s P p .002 w .38113 .19441 m .3626 .18504 L .36712 .17936 L p .162 .413 .854 r F P s P p .002 w .47833 .24109 m .49565 .24821 L .50206 .24912 L p .78 .305 .106 r F P s P p .002 w .51624 .25404 m .50206 .24912 L .49565 .24821 L p .672 .129 0 r F P s P p .002 w .50206 .24912 m .51624 .25404 L .52097 .25438 L p .723 .283 .205 r F P s P p .002 w .64179 .21382 m .62336 .22278 L .62273 .21991 L p .819 1 .83 r F P s P p .002 w .62273 .21991 m .64121 .21029 L .64179 .21382 L p .819 1 .83 r F P s P p .002 w .36039 .36546 m .37937 .35572 L .39137 .35859 L p .868 .948 .864 r F P s P p .002 w .40864 .34867 m .39137 .35859 L .37937 .35572 L p .866 .948 .865 r F P s P p .002 w .63135 .33934 m .61466 .32955 L .62408 .32759 L p .354 .502 .867 r F P s P p .002 w .62408 .32759 m .6428 .33707 L .63135 .33934 L p .354 .502 .867 r F P s P p .002 w .45556 .231 m .47426 .23949 L .47833 .24109 L p .531 0 0 r F P s P p .002 w .49565 .24821 m .47833 .24109 L .47426 .23949 L p .455 0 0 r F P s P p .002 w .36712 .17936 m .38536 .18948 L .38113 .19441 L p .162 .413 .854 r F P s P p .002 w .39969 .20374 m .38113 .19441 L .38536 .18948 L p .164 .416 .856 r F P s P p .002 w .57026 .31186 m .55709 .30316 L .56432 .30243 L p .531 .589 .858 r F P s P p .002 w .56432 .30243 m .58115 .3108 L .57026 .31186 L p .531 .589 .858 r F P s P p .002 w .53288 .29659 m .53522 .30394 L .52292 .3039 L p .694 .693 .844 r F P s P p .002 w .53522 .30394 m .53288 .29659 L .53881 .29645 L p .656 .667 .848 r F P s P p .002 w .53881 .29645 m .54703 .30368 L .53522 .30394 L p .656 .667 .848 r F P s P p .002 w .52671 .29657 m .52292 .3039 L .51143 .30358 L p .727 .728 .85 r F P s P p .002 w .52292 .3039 m .52671 .29657 L .53288 .29659 L p .694 .693 .844 r F P s P p .002 w .37937 .35572 m .39833 .34606 L .40864 .34867 L p .866 .948 .865 r F P s P p .002 w .42595 .33883 m .40864 .34867 L .39833 .34606 L p .865 .949 .867 r F P s P p .002 w .62336 .22278 m .60489 .23153 L .60427 .22929 L p .815 .999 .836 r F P s P p .002 w .60427 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.002 w .53052 .25543 m .52672 .25457 L .52097 .25438 L closepath p .299 0 0 r F P s P p .002 w .52094 .2964 m .51143 .30358 L .50195 .303 L p .759 .776 .867 r F P s P p .002 w .51143 .30358 m .52094 .2964 L .52671 .29657 L p .727 .728 .85 r F P s P p .002 w .56419 .24844 m .5474 .25371 L .54918 .2532 L p 0 0 0 r F P s P p .002 w .54918 .2532 m .56779 .24742 L .56419 .24844 L p 0 0 0 r F P s P p .002 w .39833 .34606 m .41727 .3365 L .42595 .33883 L p .865 .949 .867 r F P s P p .002 w .44329 .32911 m .42595 .33883 L .41727 .3365 L p .863 .949 .869 r F P s P p .002 w .53052 .25543 m .5438 .25414 L .53878 .25444 L closepath p .047 0 0 r F P s P p .002 w .40359 .19953 m .42179 .20946 L .41828 .21298 L p .167 .419 .858 r F P s P p .002 w .4369 .2221 m .41828 .21298 L .42179 .20946 L p .17 .424 .86 r F P s P p .002 w .53052 .25543 m .52097 .25438 L .51624 .25404 L closepath p .213 0 0 r F P s P p .002 w .60489 .23153 m .58636 .23988 L .58582 .23824 L p .808 .998 .847 r F P s P p .002 w .58582 .23824 m .60427 .22929 L .60489 .23153 L p .808 .998 .847 r F P s P p .002 w .47426 .23949 m .49298 .24716 L .49565 .24821 L p .455 0 0 r F P s P p .002 w .49565 .24821 m .51305 .25359 L .51624 .25404 L p .672 .129 0 r F P s P p .002 w .51305 .25359 m .49565 .24821 L .49298 .24716 L p 0 .276 .7 r F P s P p .002 w .59793 .31998 m .58115 .3108 L .58665 .30954 L p .358 .516 .878 r F P s P p .002 w .58665 .30954 m .60536 .31835 L .59793 .31998 L p .358 .516 .878 r F P s P p .002 w .42179 .20946 m .43997 .21924 L .4369 .2221 L p .17 .424 .86 r F P s P p .002 w .45556 .231 m .4369 .2221 L .43997 .21924 L p .176 .432 .865 r F P s P p .002 w .53052 .25543 m .5474 .25371 L .5438 .25414 L closepath p .2 .727 .751 r F P s P p .002 w .41727 .3365 m .43619 .32709 L .44329 .32911 L p .863 .949 .869 r F P s P p .002 w .46066 .31961 m .44329 .32911 L .43619 .32709 L p .858 .95 .873 r F P s P p .002 w .55709 .30316 m .54385 .29619 L .54745 .29581 L p .53 .623 .888 r F P s P p .002 w .54745 .29581 m .56432 .30243 L .55709 .30316 L p .53 .623 .888 r F P s P p .002 w .53052 .25543 m .51624 .25404 L .51305 .25359 L closepath p 0 .535 .8 r F P s P p .002 w .58636 .23988 m .56779 .24742 L .56737 .24635 L p .79 .995 .869 r F P s P p .002 w .56737 .24635 m .58582 .23824 L .58636 .23988 L p .79 .995 .869 r F P s P p .002 w .47807 .31051 m .49552 .30223 L .50195 .303 L p .813 .834 .866 r F P s P p .002 w .5162 .29611 m .50195 .303 L .49552 .30223 L p .788 .848 .896 r F P s P p .002 w .50195 .303 m .5162 .29611 L .52094 .2964 L p .759 .776 .867 r F P s P p .002 w .43997 .21924 m .45813 .22877 L .45556 .231 L p .176 .432 .865 r F P s P p .002 w .47426 .23949 m .45556 .231 L .45813 .22877 L p .187 .446 .873 r F P s P p .002 w .43619 .32709 m .45509 .31794 L .46066 .31961 L p .858 .95 .873 r F P s P p .002 w .47807 .31051 m .46066 .31961 L .45509 .31794 L p .851 .95 .88 r F P s P p .002 w .58115 .3108 m .56432 .30243 L .56794 .30156 L p .364 .535 .891 r F P s P p .002 w .56794 .30156 m .58665 .30954 L .58115 .3108 L p .364 .535 .891 r F P s P p .002 w .53052 .25543 m .54918 .2532 L .5474 .25371 L closepath p .447 .875 .921 r F P s P p .002 w .49298 .24716 m .51174 .25308 L .51305 .25359 L p 0 .276 .7 r F P s P p .002 w .53052 .25543 m .51305 .25359 L .51174 .25308 L closepath p .199 .651 .945 r F P s P p .002 w .45813 .22877 m .47627 .23785 L .47426 .23949 L p .187 .446 .873 r F P s P p .002 w .49298 .24716 m .47426 .23949 L .47627 .23785 L p .209 .476 .89 r F P s P p .002 w .56779 .24742 m .54918 .2532 L .54894 .25268 L p .743 .981 .913 r F P s P p .002 w .54894 .25268 m .56737 .24635 L .56779 .24742 L p .743 .981 .913 r F P s P p .002 w .53053 .29276 m .53288 .29659 L .52671 .29657 L closepath p .645 .769 .938 r F P s P p .002 w .53053 .29276 m .53881 .29645 L .53288 .29659 L closepath p .616 .75 .942 r F P s P p .002 w .53053 .29276 m .52671 .29657 L .52094 .2964 L closepath p .665 .794 .944 r F P s P p .002 w .53053 .29276 m .54385 .29619 L .53881 .29645 L closepath p .574 .737 .953 r F P s P p .002 w .45509 .31794 m .47398 .30922 L .47807 .31051 L p .851 .95 .88 r F P s P p .002 w .49552 .30223 m .47807 .31051 L .47398 .30922 L p .834 .951 .894 r F P s P p .002 w .53053 .29276 m .52094 .2964 L .5162 .29611 L closepath p .675 .828 .957 r F P s P p .002 w .47627 .23785 m .49438 .2461 L .49298 .24716 L p .209 .476 .89 r F P s P p .002 w .51174 .25308 m .49298 .24716 L .49438 .2461 L p .258 .545 .925 r F P s P p .002 w .56432 .30243 m .54745 .29581 L .54924 .29537 L p .379 .584 .922 r F P s P p .002 w .54924 .29537 m .56794 .30156 L .56432 .30243 L p .379 .584 .922 r F P s P p .002 w .53052 .25543 m .54894 .25268 L .54918 .2532 L closepath p .599 .908 .982 r F P s P p .002 w .53053 .29276 m .54745 .29581 L .54385 .29619 L closepath p .512 .727 .97 r F P s P p .002 w .49438 .2461 m .51247 .25256 L .51174 .25308 L p .258 .545 .925 r F P s P p .002 w .53052 .25543 m .51174 .25308 L .51247 .25256 L closepath p .375 .704 .985 r F P s P p .002 w .49552 .30223 m .513 .29571 L .5162 .29611 L p .788 .848 .896 r F P s P p .002 w .53053 .29276 m .5162 .29611 L .513 .29571 L closepath p .668 .867 .975 r F P s P p .002 w .47398 .30922 m .49284 .30134 L .49552 .30223 L p .834 .951 .894 r F P s P p .002 w .513 .29571 m .49552 .30223 L .49284 .30134 L p .787 .948 .927 r F P s P p .002 w .53052 .25543 m .51953 .2517 L .5251 .25147 L closepath p .604 .739 .94 r F P s P p .002 w .53052 .25543 m .51516 .25208 L .51953 .2517 L closepath p .557 .729 .955 r F P s P p .002 w .53052 .25543 m .51247 .25256 L .51516 .25208 L closepath p .487 .721 .975 r F P s P p .002 w .53052 .25543 m .5467 .25219 L .54894 .25268 L closepath p .663 .88 .979 r F P s P p .002 w .53052 .25543 m .54269 .25178 L .5467 .25219 L closepath p .681 .84 .96 r F P s P p .002 w .53052 .25543 m .53736 .25151 L .54269 .25178 L closepath p .676 .804 .943 r F P s P p .002 w .53052 .25543 m .53127 .2514 L .53736 .25151 L closepath p .661 .775 .934 r F P s P p .002 w .53052 .25543 m .5251 .25147 L .53127 .2514 L closepath p .637 .754 .933 r F P s P p .002 w .56737 .24635 m .54894 .25268 L .5467 .25219 L p .796 .888 .911 r F P s P p .002 w .53053 .29276 m .54924 .29537 L .54745 .29581 L closepath p .414 .712 .985 r F P s P p .002 w .49972 .24513 m .51516 .25208 L .51247 .25256 L p .478 .609 .903 r F P s P p .002 w .51247 .25256 m .49438 .2461 L .49972 .24513 L p .478 .609 .903 r F P s P p .002 w .49284 .30134 m .51169 .29526 L .513 .29571 L p .787 .948 .927 r F P s P p .002 w .53053 .29276 m .513 .29571 L .51169 .29526 L closepath p .624 .903 .985 r F P s P p .002 w .58582 .23824 m .56737 .24635 L .56293 .24534 L p .829 .883 .882 r F P s P p .002 w .5467 .25219 m .56293 .24534 L .56737 .24635 L p .796 .888 .911 r F P s P p .002 w .60427 .22929 m .58582 .23824 L .57923 .23669 L p .841 .88 .869 r F P s P p .002 w .56293 .24534 m .57923 .23669 L .58582 .23824 L p .829 .883 .882 r F P s P p .002 w .48422 .23636 m .49972 .24513 L .49438 .2461 L p .471 .57 .873 r F P s P p .002 w .49438 .2461 m .47627 .23785 L .48422 .23636 L p .471 .57 .873 r F P s P p .002 w .62273 .21991 m .60427 .22929 L .59558 .22717 L p .847 .878 .863 r F P s P p .002 w .57923 .23669 m .59558 .22717 L .60427 .22929 L p .841 .88 .869 r F P s P p .002 w .60536 .31835 m .58665 .30954 L .5861 .30821 L p 0 .271 .764 r F P s P p .002 w .58665 .30954 m .56794 .30156 L .56752 .30065 L p 0 .32 .796 r F P s P p .002 w .56752 .30065 m .5861 .30821 L .58665 .30954 L p 0 .32 .796 r F P s P p .002 w .56794 .30156 m .54924 .29537 L .549 .2949 L p .001 .434 .862 r F P s P p .002 w .549 .2949 m .56752 .30065 L .56794 .30156 L p .001 .434 .862 r F P s P p .002 w .56293 .24534 m .5467 .25219 L .54269 .25178 L p .774 .804 .877 r F P s P p .002 w .62408 .32759 m .60536 .31835 L .60474 .31663 L p 0 .247 .749 r F P s P p .002 w .5861 .30821 m .60474 .31663 L .60536 .31835 L p 0 .271 .764 r F P s P p .002 w .53053 .29276 m .549 .2949 L .54924 .29537 L closepath p .255 .671 .965 r F P s P p .002 w .6428 .33707 m .62408 .32759 L .62345 .3255 L p 0 .234 .74 r F P s P p .002 w .60474 .31663 m .62345 .3255 L .62408 .32759 L p 0 .247 .749 r F P s P p .002 w .64121 .21029 m .62273 .21991 L .61198 .21719 L p .85 .877 .86 r F P s P p .002 w .59558 .22717 m .61198 .21719 L .62273 .21991 L p .847 .878 .863 r F P s P p .002 w .66153 .3467 m .6428 .33707 L .64222 .33465 L p 0 .227 .735 r F P s P p .002 w .62345 .3255 m .64222 .33465 L .6428 .33707 L p 0 .234 .74 r F P s P p .002 w .50846 .24436 m .51953 .2517 L .51516 .25208 L p .578 .634 .87 r F P s P p .002 w .51516 .25208 m .49972 .24513 L .50846 .24436 L p .578 .634 .87 r F P s P p .002 w .68027 .35642 m .66153 .3467 L .66106 .34398 L p 0 .222 .732 r F P s P p .002 w .64222 .33465 m .66106 .34398 L .66153 .3467 L p 0 .227 .735 r F P s P p .002 w .65969 .20051 m .64121 .21029 L .62844 .20694 L p .851 .877 .858 r F P s P p .002 w .61198 .21719 m .62844 .20694 L .64121 .21029 L p .85 .877 .86 r F P s P p .002 w .49424 .30044 m .51242 .2948 L .51169 .29526 L p .565 .93 .786 r F P s P p .002 w .51169 .29526 m .49284 .30134 L .49424 .30044 L p .565 .93 .786 r F P s P p .002 w .53053 .29276 m .51169 .29526 L .51242 .2948 L closepath p .503 .895 .948 r F P s P p .002 w .699 .36622 m .68027 .35642 L .67995 .35343 L p 0 .219 .73 r F P s P p .002 w .66106 .34398 m .67995 .35343 L .68027 .35642 L p 0 .222 .732 r F P s P p .002 w .53053 .29276 m .5195 .29405 L .52509 .29384 L closepath p .198 0 0 r F P s P p .002 w .53053 .29276 m .51242 .2948 L .51511 .29438 L closepath p .274 .779 .802 r F P s P p .002 w .53053 .29276 m .51511 .29438 L .5195 .29405 L closepath p 0 0 0 r F P s P p .002 w .53053 .29276 m .54675 .29447 L .549 .2949 L closepath p .028 .566 .848 r F P s P p .002 w .53053 .29276 m .54273 .29412 L .54675 .29447 L closepath p .195 0 0 r F P s P p .002 w .53053 .29276 m .53738 .29388 L .54273 .29412 L closepath p .323 0 0 r F P s P p .002 w .53053 .29276 m .53127 .29378 L .53738 .29388 L closepath p .355 0 0 r F P s P p .002 w .53053 .29276 m .52509 .29384 L .53127 .29378 L closepath p .313 0 0 r F P s P p .002 w .46866 .22672 m .48422 .23636 L .47627 .23785 L p .468 .555 .861 r F P s P p .002 w .47627 .23785 m .45813 .22877 L .46866 .22672 L p .468 .555 .861 r F P s P p .002 w .71775 .37606 m .699 .36622 L .69892 .36299 L p 0 .216 .728 r F P s P p .002 w .67995 .35343 m .69892 .36299 L .699 .36622 L p 0 .219 .73 r F P s P p .002 w .47599 .3079 m .49424 .30044 L .49284 .30134 L p .572 .91 .683 r F P s P p .002 w .49284 .30134 m .47398 .30922 L .47599 .3079 L p .572 .91 .683 r F P s P p .002 w .67819 .19064 m .65969 .20051 L .64496 .1965 L p .852 .876 .857 r F P s P p .002 w .62844 .20694 m .64496 .1965 L .65969 .20051 L p .851 .877 .858 r F P s P p .002 w .7365 .38594 m .71775 .37606 L .71794 .37262 L p 0 .215 .727 r F P s P p .002 w .69892 .36299 m .71794 .37262 L .71775 .37606 L p 0 .216 .728 r F P s P p .002 w .75526 .39585 m .7365 .38594 L .73704 .38232 L p 0 .214 .727 r F P s P p .002 w .71794 .37262 m .73704 .38232 L .7365 .38594 L p 0 .215 .727 r F P s P p .002 w .69669 .1807 m .67819 .19064 L .66154 .18594 L p .853 .876 .856 r F P s P p .002 w .64496 .1965 m .66154 .18594 L .67819 .19064 L p .852 .876 .857 r F P s P p .002 w .77402 .40579 m .75526 .39585 L .7562 .39208 L p 0 .213 .726 r F P s P p .002 w .73704 .38232 m .7562 .39208 L .75526 .39585 L p 0 .214 .727 r F P s P p .002 w .45767 .31623 m .47599 .3079 L .47398 .30922 L p .571 .895 .635 r F P s P p .002 w .47398 .30922 m .45509 .31794 L .45767 .31623 L p .571 .895 .635 r F P s P p .002 w .79278 .41574 m .77402 .40579 L .77542 .40189 L p 0 .212 .725 r F P s P p .002 w .7562 .39208 m .77542 .40189 L .77402 .40579 L p 0 .213 .726 r F P s P p .002 w .54269 .25178 m .55494 .24452 L .56293 .24534 L p .774 .804 .877 r F P s P p .002 w .55494 .24452 m .54269 .25178 L .53736 .25151 L p .743 .747 .855 r F P s P p .002 w .45304 .21661 m .46866 .22672 L .45813 .22877 L p .466 .547 .856 r F P s P p .002 w .45813 .22877 m .43997 .21924 L .45304 .21661 L p .466 .547 .856 r F P s P p .002 w .7152 .17071 m .69669 .1807 L .67817 .17527 L p .854 .876 .855 r F P s P p .002 w .66154 .18594 m .67817 .17527 L .69669 .1807 L p .853 .876 .856 r F P s P p .002 w .81156 .4257 m .79278 .41574 L .79471 .41176 L p 0 .211 .725 r F P s P p .002 w .77542 .40189 m .79471 .41176 L .79278 .41574 L p 0 .212 .725 r F P s P p .002 w .51963 .24388 m .5251 .25147 L .51953 .2517 L p .636 .654 .851 r F P s P p .002 w .51953 .2517 m .50846 .24436 L .51963 .24388 L p .636 .654 .851 r F P s P p .002 w .83034 .43568 m .81156 .4257 L .81407 .42167 L p 0 .211 .725 r F P s P p .002 w .79471 .41176 m .81407 .42167 L .81156 .4257 L p 0 .211 .725 r F P s P p .002 w .43928 .32502 m .45767 .31623 L .45509 .31794 L p .571 .887 .611 r F P s P p .002 w .45509 .31794 m .43619 .32709 L .43928 .32502 L p .571 .887 .611 r F P s P p .002 w .73372 .16067 m .7152 .17071 L .69485 .16451 L p .854 .876 .855 r F P s P p .002 w .67817 .17527 m .69485 .16451 L .7152 .17071 L p .854 .876 .855 r F P s P p .002 w .84912 .44567 m .83034 .43568 L .83349 .43162 L p 0 .211 .725 r F P s P p .002 w .81407 .42167 m .83349 .43162 L .83034 .43568 L p 0 .211 .725 r F P s P p .002 w .56752 .30065 m .549 .2949 L .54675 .29447 L p .566 0 0 r F P s P p .002 w .86791 .45568 m .84912 .44567 L .85298 .44162 L p 0 .21 .724 r F P s P p .002 w .83349 .43162 m .85298 .44162 L .84912 .44567 L p 0 .211 .725 r F P s P p .002 w .75225 .15061 m .73372 .16067 L .7116 .15369 L p .854 .876 .855 r F P s P p .002 w .69485 .16451 m .7116 .15369 L .73372 .16067 L p .854 .876 .855 r F P s P p .002 w .8867 .46569 m .86791 .45568 L .87254 .45166 L p 0 .21 .724 r F P s P p .002 w .85298 .44162 m .87254 .45166 L .86791 .45568 L p 0 .21 .724 r F P s P p .002 w .43735 .20622 m .45304 .21661 L .43997 .21924 L p .465 .543 .852 r F P s P p .002 w .43997 .21924 m .42179 .20946 L .43735 .20622 L p .465 .543 .852 r F P s P p .002 w .42081 .33409 m .43928 .32502 L .43619 .32709 L p .571 .882 .599 r F P s P p .002 w .43619 .32709 m .41727 .3365 L .42081 .33409 L p .571 .882 .599 r F P s P p .002 w .53736 .25151 m .54424 .24396 L .55494 .24452 L p .743 .747 .855 r F P s P p .002 w .54424 .24396 m .53736 .25151 L .53127 .2514 L p .711 .707 .844 r F P s P p .002 w .57923 .23669 m .56293 .24534 L .55494 .24452 L p .792 .788 .849 r F P s P p .002 w .4996 .29962 m .51511 .29438 L .51242 .2948 L p .051 0 0 r F P s P p .002 w .51242 .2948 m .49424 .30044 L .4996 .29962 L p .051 0 0 r F P s P p .002 w .87254 .45166 m .89217 .46174 L .8867 .46569 L p 0 .21 .724 r F P s P p .002 w .77079 .14052 m .75225 .15061 L .7284 .1428 L p .854 .876 .854 r F P s P p .002 w .7116 .15369 m .7284 .1428 L .75225 .15061 L p .854 .876 .855 r F P s P p .002 w .53202 .24374 m .53127 .2514 L .5251 .25147 L p .677 .677 .843 r F P s P p .002 w .5251 .25147 m .51963 .24388 L .53202 .24374 L p .677 .677 .843 r F P s P p .002 w .53127 .2514 m .53202 .24374 L .54424 .24396 L p .711 .707 .844 r F P s P p .002 w .78934 .13041 m .77079 .14052 L .74526 .13186 L p .855 .876 .854 r F P s P p .002 w .7284 .1428 m .74526 .13186 L .77079 .14052 L p .854 .876 .854 r F P s P p .002 w .40226 .34334 m .42081 .33409 L .41727 .3365 L p .571 .879 .591 r F P s P p .002 w .41727 .3365 m .39833 .34606 L .40226 .34334 L p .571 .879 .591 r F P s P p .002 w .42159 .19563 m .43735 .20622 L .42179 .20946 L p .464 .541 .851 r F P s P p .002 w .42179 .20946 m .40359 .19953 L .42159 .19563 L p .464 .541 .851 r F P s P p .002 w .49731 .23517 m .50846 .24436 L .49972 .24513 L p .578 .605 .843 r F P s P p .002 w .49972 .24513 m .48422 .23636 L .49731 .23517 L p .578 .605 .843 r F P s P p .002 w .8079 .12029 m .78934 .13041 L .76218 .12086 L p .855 .876 .854 r F P s P p .002 w .74526 .13186 m .76218 .12086 L .78934 .13041 L p .855 .876 .854 r F P s P p .002 w .82647 .11014 m .8079 .12029 L .77915 .10981 L p .855 .876 .854 r F P s P p .002 w .76218 .12086 m .77915 .10981 L .8079 .12029 L p .855 .876 .854 r F P s P p .002 w .38364 .35273 m .40226 .34334 L .39833 .34606 L p .571 .878 .586 r F P s P p .002 w .39833 .34606 m .37937 .35572 L .38364 .35273 L p .571 .878 .586 r F P s P p .002 w .5861 .30821 m .56752 .30065 L .56307 .2998 L p .719 .185 0 r F P s P p .002 w .54675 .29447 m .56307 .2998 L .56752 .30065 L p .566 0 0 r F P s P p .002 w .56307 .2998 m .54675 .29447 L .54273 .29412 L p .725 .249 .115 r F P s P p .002 w .84504 .09999 m .82647 .11014 L .79619 .09871 L p .855 .876 .854 r F P s P p .002 w .77915 .10981 m .79619 .09871 L .82647 .11014 L p .855 .876 .854 r F P s P p .002 w .40578 .18491 m .42159 .19563 L .40359 .19953 L p .464 .54 .849 r F P s P p .002 w .40359 .19953 m .38536 .18948 L .40578 .18491 L p .464 .54 .849 r F P s P p .002 w .86363 .08982 m .84504 .09999 L .81328 .08756 L p .855 .876 .854 r F P s P p .002 w .79619 .09871 m .81328 .08756 L .84504 .09999 L p .855 .876 .854 r F P s P p .002 w .50837 .29896 m .5195 .29405 L .51511 .29438 L p .403 .009 .174 r F P s P p .002 w .51511 .29438 m .4996 .29962 L .50837 .29896 L p .403 .009 .174 r F P s P p .002 w .36494 .36223 m .38364 .35273 L .37937 .35572 L p .571 .877 .583 r F P s P p .002 w .37937 .35572 m .36039 .36546 L .36494 .36223 L p .571 .877 .583 r F P s P p .002 w .88223 .07964 m .86363 .08982 L .83044 .07637 L p .855 .876 .854 r F P s P p .002 w .81328 .08756 m .83044 .07637 L .86363 .08982 L p .855 .876 .854 r F P s P p .002 w .48399 .30669 m .4996 .29962 L .49424 .30044 L p .146 0 .01 r F P s P p .002 w .49424 .30044 m .47599 .3079 L .48399 .30669 L p .146 0 .01 r F P s P p .002 w .59558 .22717 m .57923 .23669 L .56725 .23542 L p .798 .782 .838 r F P s P p .002 w .55494 .24452 m .56725 .23542 L .57923 .23669 L p .792 .788 .849 r F P s P p .002 w .3899 .17408 m .40578 .18491 L .38536 .18948 L p .463 .539 .849 r F P s P p .002 w .38536 .18948 m .36712 .17936 L .3899 .17408 L p .463 .539 .849 r F P s P p .002 w .83044 .07637 m .84765 .06514 L .88223 .07964 L p .855 .876 .854 r F P s P p .002 w 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L .48539 .34633 L p .598 .326 .476 r F P s P p .002 w .33371 .07483 m .34589 .08703 L .27691 .09619 L p .578 .581 .82 r F P s P p .002 w .27691 .09619 m .26051 .08482 L .33371 .07483 L p .578 .581 .82 r F P s P p .002 w .7609 .06483 m .77444 .05265 L .84765 .06514 L p .805 .775 .825 r F P s P p .002 w .83868 .44771 m .85687 .45785 L .89217 .46174 L p .811 .328 .033 r F P s P p .002 w .53513 .32865 m .53593 .33719 L .57997 .33778 L p .714 .432 .497 r F P s P p .002 w .6012 .15576 m .59389 .16761 L .53744 .16636 L p .724 .672 .799 r F P s P p .002 w .25351 .43676 m .27075 .42677 L .23183 .43065 L p .207 0 .156 r F P s P p .002 w .23183 .43065 m .21249 .44065 L .25351 .43676 L p .207 0 .156 r F P s P p .002 w .53824 .15433 m .53744 .16636 L .48015 .16714 L p .685 .64 .799 r F P s P p .002 w .48015 .16714 m .47434 .15521 L .53824 .15433 L p .685 .64 .799 r F P s P p .002 w .32145 .06255 m .33371 .07483 L .26051 .08482 L p .578 .581 .819 r F P s P p .002 w .26051 .08482 m .24403 .0734 L 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MathPictureEnd } bind def /Mscale { 0 1 0 1 5 -1 roll MathScale } bind def /Plain /Courier findfont def /Bold /Courier-Bold findfont def /Italic /Courier-Oblique findfont def /MathPictureStart { /Mimatrix matrix currentmatrix def gsave newpath Mleft Mbottom translate 1 -1 scale /Mtmatrix matrix currentmatrix def Plain Mfontsize scalefont setfont 0 setgray 0 setlinewidth } bind def /MathPictureEnd { grestore } bind def /MathSubStart { Momatrix Mgmatrix Mtmatrix Mleft Mbottom Mwidth Mheight 9 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 11 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def /Mleft 0 def /Mbottom 0 def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } bind def /MathSubEnd { /Mheight exch def /Mwidth exch def /Mbottom exch def /Mleft exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto 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Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get show } for } bind def /Mshoway { 3 index Mwidthcal 5 1 roll 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get [ 6 index aload length 2 add -1 roll { pop Strform stringwidth pop neg exch add 0 rmoveto } exch kshow cleartomark } for pop } bind def /Mwidthcal { [ exch { Mwidthcal1 } forall ] [ exch dup Maxlen -1 add 0 1 3 -1 roll { [ exch 2 index { 1 index Mget exch } forall pop Maxget exch } for pop ] Mreva } bind def /Mreva { [ exch aload length -1 1 {1 roll} for ] } bind def /Mget { 1 index length -1 add 1 index ge { get } { pop pop 0 } ifelse } bind def /Maxlen { [ exch { length } forall Maxget } bind def /Maxget { counttomark -1 add 1 1 3 -1 roll { pop Mmax } for exch pop } bind def /Mwidthcal1 { [ exch { Strform stringwidth pop } forall ] } bind def /Strform { /tem (x) def tem 0 3 -1 roll put tem } bind def /Mshowa1 { 2 copy add 4 1 roll sub mul sub -2 div } bind def /MathScale { Mwidth Mheight Mlp translate scale /yscale exch def /ybias exch def /xscale exch def /xbias exch def /Momatrix xscale yscale matrix scale xbias ybias matrix translate matrix concatmatrix def /Mgmatrix matrix currentmatrix def } bind def /Mlp { 3 copy Mlpfirst { Mnodistort { Mmin dup } if 4 index 2 index 2 index Mlprun 11 index 11 -1 roll 10 -4 roll Mlp1 8 index 9 -5 roll Mlp1 4 -1 roll and { exit } if 3 -1 roll pop pop } loop exch 3 1 roll 7 -3 roll pop pop pop } bind def /Mlpfirst { 3 -1 roll dup length 2 copy -2 add get aload pop pop pop 4 -2 roll -1 add get aload pop pop pop 6 -1 roll 3 -1 roll 5 -1 roll sub div 4 1 roll exch sub div } bind def /Mlprun { 2 copy 4 index 0 get dup 4 1 roll Mlprun1 3 copy 8 -2 roll 9 -1 roll { 3 copy Mlprun1 3 copy 11 -3 roll /gt Mlpminmax 8 3 roll 11 -3 roll /lt Mlpminmax 8 3 roll } forall pop pop pop pop 3 1 roll pop pop aload pop 5 -1 roll aload pop exch 6 -1 roll Mlprun2 8 2 roll 4 -1 roll Mlprun2 6 2 roll 3 -1 roll Mlprun2 4 2 roll exch Mlprun2 6 2 roll } bind def /Mlprun1 { aload pop exch 6 -1 roll 5 -1 roll mul add 4 -2 roll mul 3 -1 roll add } bind def /Mlprun2 { 2 copy add 2 div 3 1 roll exch sub } bind def /Mlpminmax { cvx 2 index 6 index 2 index exec { 7 -3 roll 4 -1 roll } if 1 index 5 index 3 -1 roll exec { 4 1 roll pop 5 -1 roll aload pop pop 4 -1 roll aload pop [ 8 -2 roll pop 5 -2 roll pop 6 -2 roll pop 5 -1 roll ] 4 1 roll pop } { pop pop pop } ifelse } bind def /Mlp1 { 5 index 3 index sub 5 index 2 index mul 1 index le 1 index 0 le or dup not { 1 index 3 index div .99999 mul 8 -1 roll pop 7 1 roll } if 8 -1 roll 2 div 7 -2 roll pop sub 5 index 6 -3 roll pop pop mul sub exch } bind def /intop 0 def /inrht 0 def /inflag 0 def /outflag 0 def /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def /Minner { outflag 1 eq { /outflag 0 def /intop 0 def /inrht 0 def } if 5 index gsave 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cos 1 index sin neg exch Checkaux 3 1 roll pop pop } bind def /Checkaux { 4 index exch 4 index mul 3 1 roll mul add 4 1 roll } bind def /Mboxrot { Mrot 90 eq { brotaux 4 2 roll } if Mrot 180 eq { 4 2 roll brotaux 4 2 roll brotaux } if Mrot 270 eq { 4 2 roll brotaux } if } bind def /brotaux { neg exch neg } bind def /Mabsproc { 0 matrix defaultmatrix dtransform idtransform dup mul exch dup mul add sqrt } bind def /Mabswid { Mabsproc setlinewidth } bind def /Mabsdash { exch [ exch { Mabsproc } forall ] exch setdash } bind def /MBeginOrig { Momatrix concat} bind def /MEndOrig { Mgmatrix setmatrix} bind def /sampledsound where { pop} { /sampledsound { exch pop exch 5 1 roll mul 4 idiv mul 2 idiv exch pop exch /Mtempproc exch def { Mtempproc pop } repeat } bind def } ifelse /g { setgray} bind def /k { setcmykcolor} bind def /m { moveto} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /P { grestore} bind def /s { stroke} bind def /MFill { 0 0 moveto Mwidth 0 lineto Mwidth Mheight lineto 0 Mheight lineto fill } bind def /MPlotRegion { 3 index Mwidth mul 2 index Mheight mul translate exch sub Mheight mul /Mheight exch def exch sub Mwidth mul /Mwidth exch def } bind def /Mcharproc { currentfile (x) readhexstring pop 0 get exch div } bind def /Mshadeproc { dup 3 1 roll { dup Mcharproc 3 1 roll } repeat 1 eq { setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /Mrectproc { 3 index 2 index moveto 2 index 3 -1 roll lineto dup 3 1 roll lineto lineto fill } bind def /Mcolorimage { 7 1 roll pop pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index 9 index Mshadeproc Mrectproc } for pop } for pop pop pop pop } bind def /Mimage { pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index Mcharproc setgray 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.1945 .2802 L .19343 .27778 L .19318 .27718 L .19247 .27537 L .1916 .27295 L .19083 .27054 L .19016 .26812 L .18957 .2657 L .18907 .26329 L .18866 .26087 L .18835 .25856 L .18833 .25846 L .18809 .25604 L .18793 .25362 L .18784 .25121 L .18784 .24879 L .18793 .24638 L .18809 .24396 L .18833 .24154 L .18835 .24144 L .18866 .23913 L .18907 .23671 L .18957 .2343 L .19016 .23188 L .19083 .22946 L .1916 .22705 L .19247 .22463 L .19318 .22282 L Mistroke .19343 .22222 L .1945 .2198 L .19567 .21739 L .19696 .21497 L .19801 .21316 L .19838 .21255 L .19992 .21014 L .20161 .20772 L .20284 .20608 L .20345 .20531 L .20546 .20289 L .20766 .20047 L .20767 .20046 L .21007 .19806 L .2125 .19584 L .21273 .19564 L .21569 .19323 L .21734 .19199 L .219 .19081 L .22217 .18876 L .22277 .18839 L .227 .18606 L .22715 .18598 L .23183 .18381 L .23243 .18356 L .23666 .18199 L .2393 .18115 L .2415 .18054 L .24633 .17946 L .25105 .17873 L .25116 .17872 L Mfstroke P p .001 w .25599 .16417 m .26082 .1641 L .26449 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.3424 .27295 L .34169 .27537 L .3409 .27778 L .34004 .2802 L .33909 .28261 L .33813 .28485 L .33805 .28503 L .33693 .28745 L .33571 .28986 L .3344 .29228 L .3333 .29416 L .33298 .29469 L .33145 .29711 L .32981 .29953 L .32847 .30137 L .32804 .30194 L .32613 .30436 L .32408 .30677 L .32364 .30728 L .32187 .30919 L .31948 .31161 L .31881 .31226 L .31689 .31402 L .31407 .31644 L .31397 .31652 L .31098 .31885 L .30914 .32019 L .30758 .32127 L .30431 .32336 L .30377 .32369 L .29948 .32609 L .29947 .3261 L .29465 .32844 L .29447 .32852 L .28981 .33042 L Mistroke .28842 .33093 L .28498 .33207 L .28041 .33335 L .28015 .33341 L .27532 .33446 L .27049 .33521 L .26566 .33569 L .26449 .33577 L .26082 .3359 L .25599 .33583 L .25474 .33577 L .25116 .33549 L .24633 .33487 L .2415 .33397 L .2388 .33335 L .23666 .33278 L .23183 .33129 L .23081 .33093 L .227 .32947 L .22476 .32852 L .22217 .32731 L .21976 .3261 L .21734 .32478 L .21546 .32369 L .2125 .32183 L .21166 .32127 L .20825 .31885 L .20767 .31842 L .20516 .31644 L .20284 .31447 L .20234 .31402 L .19975 .31161 L .19801 .30987 L .19736 .30919 L .19515 .30677 L .19318 .30446 L .1931 .30436 L .19119 .30194 L .18943 .29953 L .18835 .29796 L .18778 .29711 L .18625 .29469 L .18484 .29228 L .18352 .28986 L .18351 .28985 L .1823 .28745 L .18118 .28503 L .18014 .28261 L .17919 .2802 L .17868 .2788 L Mistroke .17833 .27778 L .17754 .27537 L .17683 .27295 L .1762 .27054 L .17564 .26812 L .17516 .2657 L .17475 .26329 L .1744 .26087 L .17413 .25846 L .17392 .25604 L .17385 .25502 L .17379 .25362 L .17372 .25121 L .17372 .24879 L .17379 .24638 L .17385 .24498 L .17392 .24396 L .17413 .24154 L .1744 .23913 L .17475 .23671 L .17516 .2343 L .17564 .23188 L .1762 .22946 L .17683 .22705 L .17754 .22463 L .17833 .22222 L .17868 .2212 L .17919 .2198 L .18014 .21739 L .18118 .21497 L .1823 .21255 L .18351 .21015 L .18352 .21014 L .18484 .20772 L .18625 .20531 L .18778 .20289 L .18835 .20204 L .18943 .20047 L .19119 .19806 L .1931 .19564 L .19318 .19554 L .19515 .19323 L .19736 .19081 L .19801 .19013 L .19975 .18839 L .20234 .18598 L .20284 .18553 L .20516 .18356 L .20767 .18158 L .20825 .18115 L Mistroke .21166 .17873 L .2125 .17817 L .21546 .17631 L .21734 .17522 L .21976 .1739 L .22217 .17269 L .22476 .17148 L .227 .17053 L .23081 .16907 L .23183 .16871 L .23666 .16722 L .2388 .16665 L .2415 .16603 L .24633 .16513 L .25116 .16451 L .25474 .16423 L .25599 .16417 L Mfstroke P p .001 w .2415 .15159 m .24633 .15082 L .25116 .15029 L .25599 .15 L .26082 .14994 L .26566 .15012 L .27049 .15053 L .27532 .15117 L .28015 .15206 L .28059 .15216 L .28498 .1532 L .28973 .15457 L .28981 .1546 L .29465 .15627 L .29652 .15699 L .29948 .15822 L .30211 .1594 L .30431 .16047 L .30692 .16182 L .30914 .16305 L .31117 .16423 L .31397 .16599 L .31499 .16665 L .31846 .16907 L .31881 .16932 L .32165 .17148 L .32364 .17309 L .32459 .1739 L .32732 .17631 L .32847 .17739 L .32986 .17873 L .33223 .18115 L .3333 .1823 L .33444 .18356 L .33652 .18598 L .33813 .18797 L .33847 .18839 L .3403 .19081 L .34202 .19323 L .34296 .19463 L .34363 .19564 L .34515 .19806 L .34657 .20047 L .3478 .2027 L .3479 .20289 L .34915 .20531 L .35031 .20772 L .3514 .21014 L .35241 .21255 L .35263 .2131 L Mistroke .35335 .21497 L .35422 .21739 L .35502 .2198 L .35575 .22222 L .35641 .22463 L .35701 .22705 L .35746 .22903 L .35755 .22946 L .35803 .23188 L .35844 .2343 L .3588 .23671 L .35909 .23913 L .35932 .24154 L .3595 .24396 L .35962 .24638 L .35967 .24879 L .35967 .25121 L .35962 .25362 L .3595 .25604 L .35932 .25846 L .35909 .26087 L .3588 .26329 L .35844 .2657 L .35803 .26812 L .35755 .27054 L .35746 .27097 L .35701 .27295 L .35641 .27537 L .35575 .27778 L .35502 .2802 L .35422 .28261 L .35335 .28503 L .35263 .2869 L .35241 .28745 L .3514 .28986 L .35031 .29228 L .34915 .29469 L .3479 .29711 L .3478 .2973 L .34657 .29953 L .34515 .30194 L .34363 .30436 L .34296 .30537 L .34202 .30677 L .3403 .30919 L .33847 .31161 L .33813 .31203 L .33652 .31402 L .33444 .31644 L .3333 .3177 L Mistroke .33223 .31885 L .32986 .32127 L .32847 .32261 L .32732 .32369 L .32459 .3261 L .32364 .32691 L .32165 .32852 L .31881 .33068 L .31846 .33093 L .31499 .33335 L .31397 .33401 L .31117 .33577 L .30914 .33695 L .30692 .33818 L .30431 .33953 L .30211 .3406 L .29948 .34178 L .29652 .34301 L .29465 .34373 L .28981 .3454 L .28973 .34543 L .28498 .3468 L .28059 .34784 L .28015 .34794 L .27532 .34883 L .27049 .34947 L .26566 .34988 L .26082 .35006 L .25599 .35 L .25116 .34971 L .24633 .34918 L .2415 .34841 L .23863 .34784 L .23666 .3474 L .23183 .34613 L .2295 .34543 L .227 .3446 L .22271 .34301 L .22217 .3428 L .21734 .3407 L .21712 .3406 L .2125 .33828 L .21231 .33818 L .20806 .33577 L .20767 .33553 L .20424 .33335 L .20284 .3324 L .20077 .33093 L .19801 .32885 L .19758 .32852 L Mistroke .19464 .3261 L .19318 .32483 L .19191 .32369 L .18937 .32127 L .18835 .32024 L .187 .31885 L .18479 .31644 L .18351 .31498 L .18271 .31402 L .18076 .31161 L .17893 .30919 L .17868 .30885 L .17721 .30677 L .1756 .30436 L .17409 .30194 L .17385 .30155 L .17266 .29953 L .17133 .29711 L .17008 .29469 L .16902 .29249 L .16892 .29228 L .16783 .28986 L .16682 .28745 L .16588 .28503 L .16501 .28261 L .16421 .2802 L .16419 .28011 L .16348 .27778 L .16282 .27537 L .16222 .27295 L .16168 .27054 L .1612 .26812 L .16079 .2657 L .16043 .26329 L .16014 .26087 L .15991 .25846 L .15973 .25604 L .15961 .25362 L .15956 .25121 L .15956 .24879 L .15961 .24638 L .15973 .24396 L .15991 .24154 L .16014 .23913 L .16043 .23671 L .16079 .2343 L .1612 .23188 L .16168 .22946 L .16222 .22705 L .16282 .22463 L Mistroke .16348 .22222 L .16419 .21989 L .16421 .2198 L .16501 .21739 L .16588 .21497 L .16682 .21255 L .16783 .21014 L .16892 .20772 L .16902 .20751 L .17008 .20531 L .17133 .20289 L .17266 .20047 L .17385 .19845 L .17409 .19806 L .1756 .19564 L .17721 .19323 L .17868 .19115 L .17893 .19081 L .18076 .18839 L .18271 .18598 L .18351 .18502 L .18479 .18356 L .187 .18115 L .18835 .17976 L .18937 .17873 L .19191 .17631 L .19318 .17517 L .19464 .1739 L .19758 .17148 L .19801 .17115 L .20077 .16907 L .20284 .1676 L .20424 .16665 L .20767 .16447 L .20806 .16423 L .21231 .16182 L .2125 .16172 L .21712 .1594 L .21734 .1593 L .22217 .1572 L .22271 .15699 L .227 .1554 L .2295 .15457 L .23183 .15387 L .23666 .1526 L .23863 .15216 L .2415 .15159 L Mfstroke P p .001 w .25599 .13501 m .26082 .13495 L .26566 .13511 L .26833 .13524 L .27049 .13546 L .27532 .13602 L .28015 .1368 L .28445 .13766 L .28498 .13778 L .28981 .13898 L .29358 .14008 L .29465 .14041 L .29948 .14207 L .30059 .14249 L .30431 .14398 L .30644 .14491 L .30914 .14615 L .31152 .14732 L .31397 .1486 L .31605 .14974 L .31881 .15134 L .32014 .15216 L .32364 .15441 L .32388 .15457 L .32733 .15699 L .32847 .15783 L .33053 .1594 L .3333 .16164 L .33351 .16182 L .3363 .16423 L .33813 .16591 L .33892 .16665 L .34139 .16907 L .34296 .17069 L .34371 .17148 L .3459 .1739 L .3478 .1761 L .34797 .17631 L .34993 .17873 L .35179 .18115 L .35263 .18228 L .35355 .18356 L .35521 .18598 L .35678 .18839 L .35746 .18948 L .35827 .19081 L .35968 .19323 L .36102 .19564 L .36227 .19806 L .36229 .19809 L Mistroke .36346 .20047 L .36458 .20289 L .36563 .20531 L .36662 .20772 L .36712 .20903 L .36754 .21014 L .3684 .21255 L .3692 .21497 L .36995 .21739 L .37063 .2198 L .37126 .22222 L .37184 .22463 L .37196 .22517 L .37235 .22705 L .37282 .22946 L .37323 .23188 L .37359 .2343 L .3739 .23671 L .37415 .23913 L .37436 .24154 L .37451 .24396 L .37461 .24638 L .37466 .24879 L .37466 .25121 L .37461 .25362 L .37451 .25604 L .37436 .25846 L .37415 .26087 L .3739 .26329 L .37359 .2657 L .37323 .26812 L .37282 .27054 L .37235 .27295 L .37196 .27483 L .37184 .27537 L .37126 .27778 L .37063 .2802 L .36995 .28261 L .3692 .28503 L .3684 .28745 L .36754 .28986 L .36712 .29097 L .36662 .29228 L .36563 .29469 L .36458 .29711 L .36346 .29953 L .36229 .30191 L .36227 .30194 L .36102 .30436 L .35968 .30677 L Mistroke .35827 .30919 L .35746 .31052 L .35678 .31161 L .35521 .31402 L .35355 .31644 L .35263 .31772 L .35179 .31885 L .34993 .32127 L .34797 .32369 L .3478 .3239 L .3459 .3261 L .34371 .32852 L .34296 .32931 L .34139 .33093 L .33892 .33335 L .33813 .33409 L .3363 .33577 L .33351 .33818 L .3333 .33836 L .33053 .3406 L .32847 .34217 L .32733 .34301 L .32388 .34543 L .32364 .34559 L .32014 .34784 L .31881 .34866 L .31605 .35026 L .31397 .3514 L .31152 .35268 L .30914 .35385 L .30644 .35509 L .30431 .35602 L .30059 .35751 L .29948 .35793 L .29465 .35959 L .29358 .35992 L .28981 .36102 L .28498 .36222 L .28445 .36234 L .28015 .3632 L .27532 .36398 L .27049 .36454 L .26833 .36476 L .26566 .36489 L .26082 .36505 L .25599 .36499 L .25138 .36476 L .25116 .36474 L .24633 .36428 L .2415 .36362 L Mistroke .23666 .36274 L .23478 .36234 L .23183 .36165 L .227 .36033 L .22565 .35992 L .22217 .35879 L .21864 .35751 L .21734 .357 L .21279 .35509 L .2125 .35496 L .20771 .35268 L .20767 .35266 L .20318 .35026 L .20284 .35007 L .19909 .34784 L .19801 .34717 L .19535 .34543 L .19318 .34393 L .1919 .34301 L .1887 .3406 L .18835 .34032 L .18572 .33818 L .18351 .33629 L .18293 .33577 L .18031 .33335 L .17868 .33177 L .17784 .33093 L .17552 .32852 L .17385 .32669 L .17333 .3261 L .17126 .32369 L .1693 .32127 L .16902 .32092 L .16744 .31885 L .16569 .31644 L .16419 .31427 L .16402 .31402 L .16245 .31161 L .16096 .30919 L .15955 .30677 L .15935 .30643 L .15821 .30436 L .15696 .30194 L .15577 .29953 L .15465 .29711 L .15452 .29682 L .1536 .29469 L .15261 .29228 L .15169 .28986 L .15083 .28745 L Mistroke .15003 .28503 L .14969 .28396 L .14928 .28261 L .1486 .2802 L .14797 .27778 L .1474 .27537 L .14688 .27295 L .14641 .27054 L .146 .26812 L .14564 .2657 L .14533 .26329 L .14508 .26087 L .14487 .25846 L .14486 .25825 L .14472 .25604 L .14462 .25362 L .14457 .25121 L .14457 .24879 L .14462 .24638 L .14472 .24396 L .14486 .24175 L .14487 .24154 L .14508 .23913 L .14533 .23671 L .14564 .2343 L .146 .23188 L .14641 .22946 L .14688 .22705 L .1474 .22463 L .14797 .22222 L .1486 .2198 L .14928 .21739 L .14969 .21604 L .15003 .21497 L .15083 .21255 L .15169 .21014 L .15261 .20772 L .1536 .20531 L .15452 .20318 L .15465 .20289 L .15577 .20047 L .15696 .19806 L .15821 .19564 L .15935 .19357 L .15955 .19323 L .16096 .19081 L .16245 .18839 L .16402 .18598 L .16419 .18573 L .16569 .18356 L Mistroke .16744 .18115 L .16902 .17908 L .1693 .17873 L .17126 .17631 L .17333 .1739 L .17385 .17331 L .17552 .17148 L .17784 .16907 L .17868 .16823 L .18031 .16665 L .18293 .16423 L .18351 .16371 L .18572 .16182 L .18835 .15968 L .1887 .1594 L .1919 .15699 L .19318 .15607 L .19535 .15457 L .19801 .15283 L .19909 .15216 L .20284 .14993 L .20318 .14974 L .20767 .14734 L .20771 .14732 L .2125 .14504 L .21279 .14491 L .21734 .143 L .21864 .14249 L .22217 .14121 L .22565 .14008 L .227 .13967 L .23183 .13835 L .23478 .13766 L .23666 .13726 L .2415 .13638 L .24633 .13572 L .25116 .13526 L .25138 .13524 L .25599 .13501 L Mfstroke P p .001 w .25599 .11817 m .26082 .11812 L .26566 .11826 L .2675 .11833 L .27049 .11857 L .27532 .11906 L .28015 .11973 L .28498 .12058 L .28583 .12075 L .28981 .12162 L .29465 .12286 L .29575 .12316 L .29948 .12429 L .30335 .12558 L .30431 .12592 L .30914 .12777 L .30969 .128 L .31397 .12984 L .31522 .13041 L .31881 .13215 L .32014 .13283 L .32364 .1347 L .32461 .13524 L .32847 .13752 L .3287 .13766 L .33248 .14008 L .3333 .14062 L .336 .14249 L .33813 .14404 L .33929 .14491 L .34238 .14732 L .34296 .1478 L .34529 .14974 L .3478 .15193 L .34804 .15216 L .35064 .15457 L .35263 .1565 L .35311 .15699 L .35545 .1594 L .35746 .16157 L .35768 .16182 L .3598 .16423 L .36182 .16665 L .36229 .16723 L .36374 .16907 L .36558 .17148 L .36712 .17361 L .36732 .1739 L .36899 .17631 L .37058 .17873 L Mistroke .37196 .18092 L .3721 .18115 L .37354 .18356 L .37491 .18598 L .37622 .18839 L .37679 .18947 L .37747 .19081 L .37865 .19323 L .37977 .19564 L .38084 .19806 L .38162 .19992 L .38184 .20047 L .3828 .20289 L .38369 .20531 L .38454 .20772 L .38533 .21014 L .38607 .21255 L .38645 .21387 L .38676 .21497 L .3874 .21739 L .38799 .2198 L .38854 .22222 L .38903 .22463 L .38948 .22705 L .38989 .22946 L .39025 .23188 L .39056 .2343 L .39083 .23671 L .39105 .23913 L .39123 .24154 L .39128 .24246 L .39136 .24396 L .39145 .24638 L .39149 .24879 L .39149 .25121 L .39145 .25362 L .39136 .25604 L .39128 .25754 L .39123 .25846 L .39105 .26087 L .39083 .26329 L .39056 .2657 L .39025 .26812 L .38989 .27054 L .38948 .27295 L .38903 .27537 L .38854 .27778 L .38799 .2802 L .3874 .28261 L .38676 .28503 L Mistroke .38645 .28613 L .38607 .28745 L .38533 .28986 L .38454 .29228 L .38369 .29469 L .3828 .29711 L .38184 .29953 L .38162 .30008 L .38084 .30194 L .37977 .30436 L .37865 .30677 L .37747 .30919 L .37679 .31053 L .37622 .31161 L .37491 .31402 L .37354 .31644 L .3721 .31885 L .37196 .31908 L .37058 .32127 L .36899 .32369 L .36732 .3261 L .36712 .32639 L .36558 .32852 L .36374 .33093 L .36229 .33277 L .36182 .33335 L .3598 .33577 L .35768 .33818 L .35746 .33843 L .35545 .3406 L .35311 .34301 L .35263 .3435 L .35064 .34543 L .34804 .34784 L .3478 .34807 L .34529 .35026 L .34296 .3522 L .34238 .35268 L .33929 .35509 L .33813 .35596 L .336 .35751 L .3333 .35938 L .33248 .35992 L .3287 .36234 L .32847 .36248 L .32461 .36476 L .32364 .3653 L .32014 .36717 L .31881 .36785 L .31522 .36959 L Mistroke .31397 .37016 L .30969 .372 L .30914 .37223 L .30431 .37408 L .30335 .37442 L .29948 .37571 L .29575 .37684 L .29465 .37714 L .28981 .37838 L .28583 .37925 L .28498 .37942 L .28015 .38027 L .27532 .38094 L .27049 .38143 L .2675 .38167 L .26566 .38174 L .26082 .38188 L .25599 .38183 L .25211 .38167 L .25116 .38161 L .24633 .38121 L .2415 .38063 L .23666 .37987 L .2334 .37925 L .23183 .37892 L .227 .37779 L .22348 .37684 L .22217 .37645 L .21734 .37492 L .21588 .37442 L .2125 .37318 L .20954 .372 L .20767 .37122 L .20401 .36959 L .20284 .36904 L .19909 .36717 L .19801 .36661 L .19462 .36476 L .19318 .36392 L .19053 .36234 L .18835 .36097 L .18675 .35992 L .18351 .35771 L .18323 .35751 L .17994 .35509 L .17868 .35413 L .17685 .35268 L .17394 .35026 L .17385 .35019 L .17119 .34784 L Mistroke .16902 .34584 L .16859 .34543 L .16612 .34301 L .16419 .34103 L .16378 .3406 L .16155 .33818 L .15943 .33577 L .15935 .33568 L .15741 .33335 L .15549 .33093 L .15452 .32968 L .15365 .32852 L .15191 .3261 L .15024 .32369 L .14969 .32287 L .14865 .32127 L .14713 .31885 L .14569 .31644 L .14486 .31499 L .14432 .31402 L .14301 .31161 L .14176 .30919 L .14058 .30677 L .14003 .3056 L .13946 .30436 L .13839 .30194 L .13739 .29953 L .13644 .29711 L .13554 .29469 L .1352 .29373 L .13469 .29228 L .1339 .28986 L .13316 .28745 L .13247 .28503 L .13183 .28261 L .13124 .2802 L .13069 .27778 L .13036 .27621 L .1302 .27537 L .12975 .27295 L .12934 .27054 L .12898 .26812 L .12867 .2657 L .1284 .26329 L .12818 .26087 L .12801 .25846 L .12787 .25604 L .12778 .25362 L .12774 .25121 L .12774 .24879 L Mistroke .12778 .24638 L .12787 .24396 L .12801 .24154 L .12818 .23913 L .1284 .23671 L .12867 .2343 L .12898 .23188 L .12934 .22946 L .12975 .22705 L .1302 .22463 L .13036 .22379 L .13069 .22222 L .13124 .2198 L .13183 .21739 L .13247 .21497 L .13316 .21255 L .1339 .21014 L .13469 .20772 L .1352 .20627 L .13554 .20531 L .13644 .20289 L .13739 .20047 L .13839 .19806 L .13946 .19564 L .14003 .1944 L .14058 .19323 L .14176 .19081 L .14301 .18839 L .14432 .18598 L .14486 .18501 L .14569 .18356 L .14713 .18115 L .14865 .17873 L .14969 .17713 L .15024 .17631 L .15191 .1739 L .15365 .17148 L .15452 .17032 L .15549 .16907 L .15741 .16665 L .15935 .16432 L .15943 .16423 L .16155 .16182 L .16378 .1594 L .16419 .15897 L .16612 .15699 L .16859 .15457 L .16902 .15416 L .17119 .15216 L .17385 .14981 L Mistroke .17394 .14974 L .17685 .14732 L .17868 .14587 L .17994 .14491 L .18323 .14249 L .18351 .14229 L .18675 .14008 L .18835 .13903 L .19053 .13766 L .19318 .13608 L .19462 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/Mpend { MathPictureEnd } bind def /Mscale { 0 1 0 1 5 -1 roll MathScale } bind def /Plain /Courier findfont def /Bold /Courier-Bold findfont def /Italic /Courier-Oblique findfont def /MathPictureStart { /Mimatrix matrix currentmatrix def gsave newpath Mleft Mbottom translate 1 -1 scale /Mtmatrix matrix currentmatrix def Plain Mfontsize scalefont setfont 0 setgray 0 setlinewidth } bind def /MathPictureEnd { grestore } bind def /MathSubStart { Momatrix Mgmatrix Mtmatrix Mleft Mbottom Mwidth Mheight 9 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 11 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def /Mleft 0 def /Mbottom 0 def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } bind def /MathSubEnd { /Mheight exch def /Mwidth exch def /Mbottom exch def /Mleft exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } bind def /Mdot { moveto 0 0 rlineto stroke } bind def /Mtetra { moveto lineto lineto lineto fill } bind def /Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke } bind def /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash } bind def /Mfstroke { stroke currentdash pop 0 setdash } bind def /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def /Mvboxa { Mfixwid { Mvboxa1 } { dup Mwidthcal 0 exch { add } forall exch Mvboxa1 4 index 7 -1 roll add 4 -1 roll pop 3 1 roll } ifelse } bind def /Mvboxa1 { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def /Mbbox { 0 0 moveto false charpath flattenpath pathbbox newpath } bind def /Mmin { 2 copy gt { exch } if pop } bind def /Mmax { 2 copy lt { exch } if pop } bind def /Mrotshowa { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix Mshowa /Mtmatrix exch def /Mrot 0 def } bind def /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mfixwid { Mshowax } { Mshoway } ifelse pop pop pop pop Mgmatrix setmatrix } bind def /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get show } for } bind def /Mshoway { 3 index Mwidthcal 5 1 roll 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get [ 6 index aload length 2 add -1 roll { pop Strform stringwidth pop neg exch add 0 rmoveto } exch kshow cleartomark } for pop } bind def /Mwidthcal { [ exch { Mwidthcal1 } forall ] [ exch dup Maxlen -1 add 0 1 3 -1 roll { [ exch 2 index { 1 index Mget exch } forall pop Maxget exch } for pop ] Mreva } bind def /Mreva { [ exch aload length -1 1 {1 roll} for ] } bind def /Mget { 1 index length -1 add 1 index ge { get } { pop pop 0 } ifelse } bind def /Maxlen { [ exch { length } forall Maxget } bind def /Maxget { counttomark -1 add 1 1 3 -1 roll { pop Mmax } for exch pop } bind def /Mwidthcal1 { [ exch { Strform stringwidth pop } forall ] } bind def /Strform { /tem (x) def tem 0 3 -1 roll put tem } bind def /Mshowa1 { 2 copy add 4 1 roll sub mul sub -2 div } bind def /MathScale { Mwidth Mheight Mlp translate scale /yscale exch def /ybias exch def /xscale exch def /xbias exch def /Momatrix xscale yscale matrix scale xbias ybias matrix translate matrix concatmatrix def /Mgmatrix matrix currentmatrix def } bind def /Mlp { 3 copy Mlpfirst { Mnodistort { Mmin dup } if 4 index 2 index 2 index Mlprun 11 index 11 -1 roll 10 -4 roll Mlp1 8 index 9 -5 roll Mlp1 4 -1 roll and { exit } if 3 -1 roll pop pop } loop exch 3 1 roll 7 -3 roll pop pop pop } bind def /Mlpfirst { 3 -1 roll dup length 2 copy -2 add get aload pop pop pop 4 -2 roll -1 add get aload pop pop pop 6 -1 roll 3 -1 roll 5 -1 roll sub div 4 1 roll exch sub div } bind def /Mlprun { 2 copy 4 index 0 get dup 4 1 roll Mlprun1 3 copy 8 -2 roll 9 -1 roll { 3 copy Mlprun1 3 copy 11 -3 roll /gt Mlpminmax 8 3 roll 11 -3 roll /lt Mlpminmax 8 3 roll } forall pop pop pop pop 3 1 roll pop pop aload pop 5 -1 roll aload pop exch 6 -1 roll Mlprun2 8 2 roll 4 -1 roll Mlprun2 6 2 roll 3 -1 roll Mlprun2 4 2 roll exch Mlprun2 6 2 roll } bind def /Mlprun1 { aload pop exch 6 -1 roll 5 -1 roll mul add 4 -2 roll mul 3 -1 roll add } bind def /Mlprun2 { 2 copy add 2 div 3 1 roll exch sub } bind def /Mlpminmax { cvx 2 index 6 index 2 index exec { 7 -3 roll 4 -1 roll } if 1 index 5 index 3 -1 roll exec { 4 1 roll pop 5 -1 roll aload pop pop 4 -1 roll aload pop [ 8 -2 roll pop 5 -2 roll pop 6 -2 roll pop 5 -1 roll ] 4 1 roll pop } { pop pop pop } ifelse } bind def /Mlp1 { 5 index 3 index sub 5 index 2 index mul 1 index le 1 index 0 le or dup not { 1 index 3 index div .99999 mul 8 -1 roll pop 7 1 roll } if 8 -1 roll 2 div 7 -2 roll pop sub 5 index 6 -3 roll pop pop mul sub exch } bind def /intop 0 def /inrht 0 def /inflag 0 def /outflag 0 def /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def /Minner { outflag 1 eq { /outflag 0 def /intop 0 def /inrht 0 def } if 5 index gsave Mtmatrix setmatrix Mvboxa pop grestore 3 -1 roll pop dup intop gt { /intop exch def } { pop } ifelse dup inrht gt { /inrht exch def } { pop } ifelse pop /inflag 1 def } bind def /Mouter { /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def inflag 1 eq { dup 0 lt { dup intop mul neg /yadtop exch def } if dup 0 gt { dup intop mul /yadbot exch def } if pop dup 0 lt { dup inrht mul neg /xadrht exch def } if dup 0 gt { dup inrht mul /xadlft exch def } if pop /outflag 1 def } { pop pop} ifelse /inflag 0 def /inrht 0 def /intop 0 def } bind def /Mboxout { outflag 1 eq { 4 -1 roll xadlft leadjust add sub 4 1 roll 3 -1 roll yadbot leadjust add sub 3 1 roll exch xadrht leadjust add add exch yadtop leadjust add add /outflag 0 def /xadlft 0 def /yadbot 0 def /xadrht 0 def /yadtop 0 def } if } bind def /leadjust { (m) stringwidth pop .5 mul } bind def /Mrotcheck { dup 90 eq { yadbot /yadbot xadrht def /xadrht yadtop def /yadtop xadlft def /xadlft exch def } if dup cos 1 index sin Checkaux dup cos 1 index sin neg exch Checkaux 3 1 roll pop pop } bind def /Checkaux { 4 index exch 4 index mul 3 1 roll mul add 4 1 roll } bind def /Mboxrot { Mrot 90 eq { brotaux 4 2 roll } if Mrot 180 eq { 4 2 roll brotaux 4 2 roll brotaux } if Mrot 270 eq { 4 2 roll brotaux } if } bind def /brotaux { neg exch neg } bind def /Mabsproc { 0 matrix defaultmatrix dtransform idtransform dup mul exch dup mul add sqrt } bind def /Mabswid { Mabsproc setlinewidth } bind def /Mabsdash { exch [ exch { Mabsproc } forall ] exch setdash } bind def /MBeginOrig { Momatrix concat} bind def /MEndOrig { Mgmatrix setmatrix} bind def /sampledsound where { pop} { /sampledsound { exch pop exch 5 1 roll mul 4 idiv mul 2 idiv exch pop exch /Mtempproc exch def { Mtempproc pop } repeat } bind def } ifelse /g { setgray} bind def /k { setcmykcolor} bind def /m { moveto} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /P { grestore} bind def /s { stroke} bind def /MFill { 0 0 moveto Mwidth 0 lineto Mwidth Mheight lineto 0 Mheight lineto fill } bind def /MPlotRegion { 3 index Mwidth mul 2 index Mheight mul translate exch sub Mheight mul /Mheight exch def exch sub Mwidth mul /Mwidth exch def } bind def /Mcharproc { currentfile (x) readhexstring pop 0 get exch div } bind def /Mshadeproc { dup 3 1 roll { dup Mcharproc 3 1 roll } repeat 1 eq { setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /Mrectproc { 3 index 2 index moveto 2 index 3 -1 roll lineto dup 3 1 roll lineto lineto fill } bind def /Mcolorimage { 7 1 roll pop pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index 9 index Mshadeproc Mrectproc } for pop } for pop pop pop pop } bind def /Mimage { pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index Mcharproc setgray Mrectproc } for pop } for pop pop pop } bind def /Mnodistort true def /colorimage {Mcolorimage} bind def /image {Mimage} bind def /Mleft 0.000000 def /Mbottom 236.000000 def /Mwidth 472.000000 def /Mheight 236.000000 def /Mfontsize 12 def /Plain /Courier findfont def 0 Mbottom Mheight neg add 2 mul Mheight add translate 1 -1 scale MathPictureStart /Courier findfont 10 scalefont setfont 0.5 0.120192 0.25 0.120192 [ [ -0.001 -0.001 0 0 ] [ 1.001 .501 0 0 ] ] MathScale 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m 0 .5 L s P P p p .002 w 0 .5 m 1 .5 L s P p .002 w 1 0 m 1 .5 L s P P p p .002 w 0 0 m 1 0 L s P p .002 w .5 0 m .5 .5 L s P P 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath p p .001 w .25599 .28641 m .26082 .28616 L .26566 .28693 L .26776 .28745 L .27049 .2888 L .27239 .28986 L .27532 .29216 L .27545 .29228 L .27771 .29469 L .2793 .29711 L .28015 .2989 L .28037 .29953 L .28099 .30194 L .28117 .30436 L .28088 .30677 L .28015 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.26986 L .80596 .27054 L .80256 .27295 L .80199 .27332 L .79854 .27537 L Mistroke .79716 .27611 L .79377 .27778 L .79233 .27844 L .78803 .2802 L .7875 .2804 L .78266 .28207 L .7809 .28261 L .77783 .28349 L .773 .28469 L .77144 .28503 L .76817 .28569 L .76334 .28651 L .7585 .28716 L .7559 .28745 L .75367 .28765 L .74884 .28798 L .74401 .28817 L .73918 .28821 L .73434 .28809 L .72951 .28783 L .7249 .28745 L .72468 .28742 L .71985 .28685 L .71502 .28612 L .71019 .28521 L .70933 .28503 L .70535 .28411 L .70052 .28281 L .69987 .28261 L .69569 .28127 L .69274 .2802 L .69086 .27946 L .687 .27778 L .68603 .27732 L .68223 .27537 L .68119 .27478 L .67821 .27295 L .67636 .27169 L .67481 .27054 L .67192 .26812 L .67153 .26776 L .66949 .2657 L .66747 .26329 L .6667 .26222 L .66584 .26087 L .66455 .25846 L .6636 .25604 L .66297 .25362 L .66266 .25121 L .66266 .24879 L Mistroke .66297 .24638 L .6636 .24396 L .66455 .24154 L .66584 .23913 L .6667 .23778 L .66747 .23671 L .66949 .2343 L .67153 .23224 L .67192 .23188 L .67481 .22946 L .67636 .22831 L .67821 .22705 L .68119 .22522 L .68223 .22463 L .68603 .22268 L .687 .22222 L .69086 .22054 L .69274 .2198 L .69569 .21873 L .69987 .21739 L .70052 .21719 L .70535 .21589 L .70933 .21497 L .71019 .21479 L .71502 .21388 L .71985 .21315 L .72468 .21258 L .7249 .21255 L .72951 .21217 L Mfstroke P p .001 w .2415 .12001 m .24633 .11941 L .25116 .11899 L .25599 .11876 L .26082 .11872 L .26566 .11885 L .27049 .11918 L .27532 .11969 L .28015 .12039 L .28225 .12075 L .28498 .12128 L .28981 .12237 L .29286 .12316 L .29465 .12367 L .29948 .12518 L .30065 .12558 L .30431 .12692 L .30702 .128 L .30914 .1289 L .31248 .13041 L .31397 .13113 L .3173 .13283 L .31881 .13365 L .32161 .13524 L .32364 .13647 L .32552 .13766 L .32847 .13965 L .32908 .14008 L .33236 .14249 L .3333 .14323 L .33537 .14491 L .33813 .1473 L .33816 .14732 L .34074 .14974 L .34296 .15199 L .34312 .15216 L .34533 .15457 L .34737 .15699 L .3478 .15751 L .34925 .1594 L .35099 .16182 L .35258 .16423 L .35263 .16432 L .35403 .16665 L .35534 .16907 L .35653 .17148 L .35746 .17358 L .35759 .1739 L .35853 .17631 L .35935 .17873 L Mistroke .36006 .18115 L .36065 .18356 L .36112 .18598 L .36149 .18839 L .36176 .19081 L .36192 .19323 L .36198 .19564 L .36194 .19806 L .36181 .20047 L .3616 .20289 L .3613 .20531 L .36093 .20772 L .36049 .21014 L .35998 .21255 L .35943 .21497 L .35882 .21739 L .35819 .2198 L .35753 .22222 L .35746 .22247 L .35686 .22463 L .3562 .22705 L .35554 .22946 L .35492 .23188 L .35434 .2343 L .35381 .23671 L .35335 .23913 L .35297 .24154 L .35267 .24396 L .35263 .24429 L .35247 .24638 L .35237 .24879 L .35237 .25121 L .35247 .25362 L .35263 .25571 L .35267 .25604 L .35297 .25846 L .35335 .26087 L .35381 .26329 L .35434 .2657 L .35492 .26812 L .35554 .27054 L .3562 .27295 L .35686 .27537 L .35746 .27753 L .35753 .27778 L .35819 .2802 L .35882 .28261 L .35943 .28503 L .35998 .28745 L .36049 .28986 L Mistroke .36093 .29228 L .3613 .29469 L .3616 .29711 L .36181 .29953 L .36194 .30194 L .36198 .30436 L .36192 .30677 L .36176 .30919 L .36149 .31161 L .36112 .31402 L .36065 .31644 L .36006 .31885 L .35935 .32127 L .35853 .32369 L .35759 .3261 L .35746 .32642 L .35653 .32852 L .35534 .33093 L .35403 .33335 L .35263 .33568 L .35258 .33577 L .35099 .33818 L .34925 .3406 L .3478 .34249 L .34737 .34301 L .34533 .34543 L .34312 .34784 L .34296 .34801 L .34074 .35026 L .33816 .35268 L .33813 .3527 L .33537 .35509 L .3333 .35677 L .33236 .35751 L .32908 .35992 L .32847 .36035 L .32552 .36234 L .32364 .36353 L .32161 .36476 L .31881 .36635 L .3173 .36717 L .31397 .36887 L .31248 .36959 L .30914 .3711 L .30702 .372 L .30431 .37308 L .30065 .37442 L .29948 .37482 L .29465 .37633 L .29286 .37684 L Mistroke .28981 .37763 L .28498 .37872 L .28225 .37925 L .28015 .37961 L .27532 .38031 L .27049 .38082 L .26566 .38115 L .26082 .38128 L .25599 .38124 L .25116 .38101 L .24633 .38059 L .2415 .37999 L .23699 .37925 L .23666 .37919 L .23183 .3782 L .227 .37701 L .22637 .37684 L .22217 .3756 L .21859 .37442 L .21734 .37398 L .2125 .37212 L .21222 .372 L .20767 .37002 L .20675 .36959 L .20284 .36765 L .20194 .36717 L .19801 .36498 L .19762 .36476 L .19372 .36234 L .19318 .36199 L .19015 .35992 L .18835 .35862 L .18687 .35751 L .18386 .35509 L .18351 .3548 L .18107 .35268 L .17868 .35044 L .17849 .35026 L .17611 .34784 L .1739 .34543 L .17385 .34537 L .17186 .34301 L .16998 .3406 L .16902 .33929 L .16824 .33818 L .16666 .33577 L .16521 .33335 L .16419 .3315 L .16389 .33093 L .1627 .32852 L Mistroke .16164 .3261 L .1607 .32369 L .15988 .32127 L .15935 .31951 L .15918 .31885 L .15859 .31644 L .15811 .31402 L .15774 .31161 L .15748 .30919 L .15732 .30677 L .15726 .30436 L .15729 .30194 L .15742 .29953 L .15763 .29711 L .15793 .29469 L .1583 .29228 L .15875 .28986 L .15925 .28745 L .15935 .28698 L .15981 .28503 L .16041 .28261 L .16105 .2802 L .1617 .27778 L .16237 .27537 L .16304 .27295 L .16369 .27054 L .16419 .26863 L .16431 .26812 L .1649 .2657 L .16542 .26329 L .16589 .26087 L .16627 .25846 L .16656 .25604 L .16677 .25362 L .16687 .25121 L .16687 .24879 L .16677 .24638 L .16656 .24396 L .16627 .24154 L .16589 .23913 L .16542 .23671 L .1649 .2343 L .16431 .23188 L .16419 .23137 L .16369 .22946 L .16304 .22705 L .16237 .22463 L .1617 .22222 L .16105 .2198 L .16041 .21739 L Mistroke .15981 .21497 L .15935 .21302 L .15925 .21255 L .15875 .21014 L .1583 .20772 L .15793 .20531 L .15763 .20289 L .15742 .20047 L .15729 .19806 L .15726 .19564 L .15732 .19323 L .15748 .19081 L .15774 .18839 L .15811 .18598 L .15859 .18356 L .15918 .18115 L .15935 .18049 L .15988 .17873 L .1607 .17631 L .16164 .1739 L .1627 .17148 L .16389 .16907 L .16419 .1685 L .16521 .16665 L .16666 .16423 L .16824 .16182 L .16902 .16071 L .16998 .1594 L .17186 .15699 L .17385 .15463 L .1739 .15457 L .17611 .15216 L .17849 .14974 L .17868 .14956 L .18107 .14732 L .18351 .1452 L .18386 .14491 L .18687 .14249 L .18835 .14138 L .19015 .14008 L .19318 .13801 L .19372 .13766 L .19762 .13524 L .19801 .13502 L .20194 .13283 L .20284 .13235 L .20675 .13041 L .20767 .12998 L .21222 .128 L .2125 .12788 L Mistroke .21734 .12602 L .21859 .12558 L .22217 .1244 L .22637 .12316 L .227 .12299 L .23183 .1218 L .23666 .12081 L .23699 .12075 L .2415 .12001 L Mfstroke P p .001 w .71502 .19786 m .71985 .19735 L .72468 .19695 L .72951 .19665 L .73434 .19647 L .73918 .19639 L .74401 .19642 L .74884 .19655 L .75367 .19679 L .7585 .19713 L .76334 .19759 L .76741 .19806 L .76817 .19815 L .773 .19884 L .77783 .19964 L .78216 .20047 L .78266 .20058 L .7875 .20164 L .79233 .20285 L .79247 .20289 L .79716 .20422 L .80064 .20531 L .80199 .20575 L .80682 .20747 L .80748 .20772 L .81165 .20941 L .81334 .21014 L .81649 .21158 L .81847 .21255 L .82132 .21405 L .82298 .21497 L .82615 .21686 L .82698 .21739 L .83054 .2198 L .83098 .22013 L .8337 .22222 L .83581 .22401 L .83651 .22463 L .83899 .22705 L .84065 .22885 L .84117 .22946 L .84307 .23188 L .84471 .2343 L .84548 .23559 L .84609 .23671 L .84722 .23913 L .84813 .24154 L .8488 .24396 L .84924 .24638 L .84946 .24879 L Mistroke .84946 .25121 L .84924 .25362 L .8488 .25604 L .84813 .25846 L .84722 .26087 L .84609 .26329 L .84548 .26441 L .84471 .2657 L .84307 .26812 L .84117 .27054 L .84065 .27115 L .83899 .27295 L .83651 .27537 L .83581 .27599 L .8337 .27778 L .83098 .27987 L .83054 .2802 L .82698 .28261 L .82615 .28314 L .82298 .28503 L .82132 .28595 L .81847 .28745 L .81649 .28842 L .81334 .28986 L .81165 .29059 L .80748 .29228 L .80682 .29253 L .80199 .29425 L .80064 .29469 L .79716 .29578 L .79247 .29711 L .79233 .29715 L .7875 .29836 L .78266 .29942 L .78216 .29953 L .77783 .30036 L .773 .30116 L .76817 .30185 L .76741 .30194 L .76334 .30241 L .7585 .30287 L .75367 .30321 L .74884 .30345 L .74401 .30358 L .73918 .30361 L .73434 .30353 L .72951 .30335 L .72468 .30305 L .71985 .30265 L .71502 .30214 L Mistroke .71336 .30194 L .71019 .30152 L .70535 .30077 L .70052 .29991 L .6986 .29953 L .69569 .29891 L .69086 .29777 L .6883 .29711 L .68603 .29649 L .68119 .29504 L .68013 .29469 L .67636 .29341 L .67329 .29228 L .67153 .29159 L .66742 .28986 L .6667 .28954 L .6623 .28745 L .66187 .28723 L .65779 .28503 L .65704 .2846 L .65378 .28261 L .6522 .28158 L .65023 .2802 L .64737 .27803 L .64706 .27778 L .64426 .27537 L .64254 .27373 L .64177 .27295 L .63959 .27054 L .63771 .26815 L .63769 .26812 L .63605 .2657 L .63467 .26329 L .63353 .26087 L .63288 .25919 L .63263 .25846 L .63196 .25604 L .63152 .25362 L .63129 .25121 L .63129 .24879 L .63152 .24638 L .63196 .24396 L .63263 .24154 L .63288 .24081 L .63353 .23913 L .63467 .23671 L .63605 .2343 L .63769 .23188 L .63771 .23185 L .63959 .22946 L Mistroke .64177 .22705 L .64254 .22627 L .64426 .22463 L .64706 .22222 L .64737 .22197 L .65023 .2198 L .6522 .21842 L .65378 .21739 L .65704 .2154 L .65779 .21497 L .66187 .21277 L .6623 .21255 L .6667 .21046 L .66742 .21014 L .67153 .20841 L .67329 .20772 L .67636 .20659 L .68013 .20531 L .68119 .20496 L .68603 .20351 L .6883 .20289 L .69086 .20223 L .69569 .20109 L .6986 .20047 L .70052 .20009 L .70535 .19923 L .71019 .19848 L .71336 .19806 L .71502 .19786 L Mfstroke P p .001 w .2415 .09875 m .24633 .09824 L .25116 .0979 L .25599 .0977 L .26082 .09766 L .26566 .09778 L .27049 .09805 L .27532 .09848 L .27973 .099 L .28015 .09906 L .28498 .09981 L .28981 .10071 L .29309 .10142 L .29465 .10179 L .29948 .10303 L .30232 .10384 L .30431 .10444 L .30914 .10604 L .30975 .10625 L .31397 .10782 L .31609 .10867 L .31881 .1098 L .32169 .11108 L .32364 .11199 L .32672 .1135 L .32847 .1144 L .3313 .11592 L .3333 .11704 L .33552 .11833 L .33813 .11993 L .33942 .12075 L .34296 .1231 L .34306 .12316 L .34645 .12558 L .3478 .12658 L .34964 .128 L .35263 .13041 L .35263 .13041 L .35545 .13283 L .35746 .13464 L .35811 .13524 L .36061 .13766 L .36229 .13936 L .36297 .14008 L .3652 .14249 L .36712 .1447 L .3673 .14491 L .36928 .14732 L .37114 .14974 L .37196 .15084 L Mistroke .37289 .15216 L .37453 .15457 L .37606 .15699 L .37679 .1582 L .37748 .1594 L .37881 .16182 L .38003 .16423 L .38115 .16665 L .38162 .16772 L .38218 .16907 L .38311 .17148 L .38395 .1739 L .38469 .17631 L .38534 .17873 L .3859 .18115 L .38637 .18356 L .38645 .18405 L .38675 .18598 L .38704 .18839 L .38726 .19081 L .38738 .19323 L .38743 .19564 L .3874 .19806 L .3873 .20047 L .38713 .20289 L .3869 .20531 L .3866 .20772 L .38645 .20878 L .38625 .21014 L .38585 .21255 L .3854 .21497 L .38493 .21739 L .38442 .2198 L .38391 .22222 L .38338 .22463 L .38286 .22705 L .38235 .22946 L .38186 .23188 L .38162 .23313 L .38141 .2343 L .381 .23671 L .38064 .23913 L .38034 .24154 L .38011 .24396 L .37996 .24638 L .37988 .24879 L .37988 .25121 L .37996 .25362 L .38011 .25604 L .38034 .25846 L Mistroke .38064 .26087 L .381 .26329 L .38141 .2657 L .38162 .26687 L .38186 .26812 L .38235 .27054 L .38286 .27295 L .38338 .27537 L .38391 .27778 L .38442 .2802 L .38493 .28261 L .3854 .28503 L .38585 .28745 L .38625 .28986 L .38645 .29122 L .3866 .29228 L .3869 .29469 L .38713 .29711 L .3873 .29953 L .3874 .30194 L .38743 .30436 L .38738 .30677 L .38726 .30919 L .38704 .31161 L .38675 .31402 L .38645 .31595 L .38637 .31644 L .3859 .31885 L .38534 .32127 L .38469 .32369 L .38395 .3261 L .38311 .32852 L .38218 .33093 L .38162 .33228 L .38115 .33335 L .38003 .33577 L .37881 .33818 L .37748 .3406 L .37679 .3418 L .37606 .34301 L .37453 .34543 L .37289 .34784 L .37196 .34916 L .37114 .35026 L .36928 .35268 L .3673 .35509 L .36712 .3553 L .3652 .35751 L .36297 .35992 L .36229 .36064 L Mistroke .36061 .36234 L .35811 .36476 L .35746 .36536 L .35545 .36717 L .35263 .36959 L .35263 .36959 L .34964 .372 L .3478 .37342 L .34645 .37442 L .34306 .37684 L .34296 .3769 L .33942 .37925 L .33813 .38007 L .33552 .38167 L .3333 .38296 L .3313 .38408 L .32847 .3856 L .32672 .3865 L .32364 .38801 L .32169 .38892 L .31881 .3902 L .31609 .39133 L .31397 .39218 L .30975 .39375 L .30914 .39396 L .30431 .39556 L .30232 .39616 L .29948 .39697 L .29465 .39821 L .29309 .39858 L .28981 .39929 L .28498 .40019 L .28015 .40094 L .27973 .401 L .27532 .40152 L .27049 .40195 L .26566 .40222 L .26082 .40234 L .25599 .4023 L .25116 .4021 L .24633 .40176 L .2415 .40125 L .23949 .401 L .23666 .40059 L .23183 .39976 L .227 .39877 L .22614 .39858 L .22217 .39762 L .21734 .39629 L .21692 .39616 L Mistroke .2125 .39478 L .20948 .39375 L .20767 .39309 L .20314 .39133 L .20284 .39121 L .19801 .38913 L .19754 .38892 L .19318 .38683 L .19251 .3865 L .18835 .38431 L .18793 .38408 L .18371 .38167 L .18351 .38155 L .17981 .37925 L .17868 .37852 L .17618 .37684 L .17385 .3752 L .17278 .37442 L .16959 .372 L .16902 .37155 L .1666 .36959 L .16419 .36753 L .16378 .36717 L .16112 .36476 L .15935 .36306 L .15862 .36234 L .15626 .35992 L .15452 .35806 L .15403 .35751 L .15193 .35509 L .14995 .35268 L .14969 .35235 L .14809 .35026 L .14634 .34784 L .14486 .34566 L .1447 .34543 L .14317 .34301 L .14175 .3406 L .14043 .33818 L .14003 .33741 L .1392 .33577 L .13808 .33335 L .13705 .33093 L .13612 .32852 L .13529 .3261 L .1352 .32581 L .13455 .32369 L .1339 .32127 L .13334 .31885 L .13287 .31644 L Mistroke .13249 .31402 L .13219 .31161 L .13198 .30919 L .13186 .30677 L .13181 .30436 L .13184 .30194 L .13194 .29953 L .13211 .29711 L .13235 .29469 L .13265 .29228 L .133 .28986 L .1334 .28745 L .13384 .28503 L .13432 .28261 L .13482 .2802 L .1352 .27845 L .13534 .27778 L .13587 .27537 L .13639 .27295 L .1369 .27054 L .13739 .26812 L .13784 .2657 L .13825 .26329 L .13861 .26087 L .1389 .25846 L .13913 .25604 L .13929 .25362 L .13937 .25121 L .13937 .24879 L .13929 .24638 L .13913 .24396 L .1389 .24154 L .13861 .23913 L .13825 .23671 L .13784 .2343 L .13739 .23188 L .1369 .22946 L .13639 .22705 L .13587 .22463 L .13534 .22222 L .1352 .22155 L .13482 .2198 L .13432 .21739 L .13384 .21497 L .1334 .21255 L .133 .21014 L .13265 .20772 L .13235 .20531 L .13211 .20289 L .13194 .20047 L Mistroke .13184 .19806 L .13181 .19564 L .13186 .19323 L .13198 .19081 L .13219 .18839 L .13249 .18598 L .13287 .18356 L .13334 .18115 L .1339 .17873 L .13455 .17631 L .1352 .17419 L .13529 .1739 L .13612 .17148 L .13705 .16907 L .13808 .16665 L .1392 .16423 L .14003 .16259 L .14043 .16182 L .14175 .1594 L .14317 .15699 L .1447 .15457 L .14486 .15434 L .14634 .15216 L .14809 .14974 L .14969 .14765 L .14995 .14732 L .15193 .14491 L .15403 .14249 L .15452 .14194 L .15626 .14008 L .15862 .13766 L .15935 .13694 L .16112 .13524 L .16378 .13283 L .16419 .13247 L .1666 .13041 L .16902 .12845 L .16959 .128 L .17278 .12558 L .17385 .1248 L .17618 .12316 L .17868 .12148 L .17981 .12075 L .18351 .11845 L .18371 .11833 L .18793 .11592 L .18835 .11569 L .19251 .1135 L .19318 .11317 L .19754 .11108 L Mistroke .19801 .11087 L .20284 .10879 L .20314 .10867 L .20767 .10691 L .20948 .10625 L .2125 .10522 L .21692 .10384 L .21734 .10371 L .22217 .10238 L .22614 .10142 L .227 .10123 L .23183 .10024 L .23666 .09941 L .23949 .099 L .2415 .09875 L Mfstroke P p .001 w .71502 .17833 m .71985 .17796 L .72468 .17766 L .72951 .17745 L .73434 .17731 L .73918 .17726 L .74401 .17728 L .74884 .17737 L .75367 .17755 L .7585 .1778 L .76334 .17813 L .76817 .17855 L .77007 .17873 L .773 .17904 L .77783 .17962 L .78266 .18028 L .7875 .18103 L .78816 .18115 L .79233 .18188 L .79716 .18282 L .80069 .18356 L .80199 .18385 L .80682 .18499 L .81068 .18598 L .81165 .18624 L .81649 .1876 L .81912 .18839 L .82132 .18909 L .82615 .1907 L .82645 .19081 L .83098 .19247 L .83295 .19323 L .83581 .19438 L .83877 .19564 L .84065 .19648 L .84404 .19806 L .84548 .19876 L .84882 .20047 L .85031 .20127 L .85319 .20289 L .85514 .20404 L .85719 .20531 L .85997 .20712 L .86086 .20772 L .86422 .21014 L .8648 .21058 L .86731 .21255 L .86964 .21453 L .87013 .21497 L .87272 .21739 L Mistroke .87447 .21916 L .87507 .2198 L .87721 .22222 L .87914 .22463 L .8793 .22485 L .88087 .22705 L .88241 .22946 L .88376 .23188 L .88413 .2326 L .88494 .2343 L .88594 .23671 L .88677 .23913 L .88743 .24154 L .88792 .24396 L .88825 .24638 L .88841 .24879 L .88841 .25121 L .88825 .25362 L .88792 .25604 L .88743 .25846 L .88677 .26087 L .88594 .26329 L .88494 .2657 L .88413 .2674 L .88376 .26812 L .88241 .27054 L .88087 .27295 L .8793 .27515 L .87914 .27537 L .87721 .27778 L .87507 .2802 L .87447 .28084 L .87272 .28261 L .87013 .28503 L .86964 .28547 L .86731 .28745 L .8648 .28942 L .86422 .28986 L .86086 .29228 L .85997 .29288 L .85719 .29469 L .85514 .29596 L .85319 .29711 L .85031 .29873 L .84882 .29953 L .84548 .30124 L .84404 .30194 L .84065 .30352 L .83877 .30436 L .83581 .30562 L Mistroke .83295 .30677 L .83098 .30753 L .82645 .30919 L .82615 .3093 L .82132 .31091 L .81912 .31161 L .81649 .3124 L .81165 .31376 L .81068 .31402 L .80682 .31501 L .80199 .31615 L .80069 .31644 L .79716 .31718 L .79233 .31812 L .78816 .31885 L .7875 .31897 L .78266 .31972 L .77783 .32038 L .773 .32096 L .77007 .32127 L .76817 .32145 L .76334 .32187 L .7585 .3222 L .75367 .32245 L .74884 .32263 L .74401 .32272 L .73918 .32274 L .73434 .32269 L .72951 .32255 L .72468 .32234 L .71985 .32204 L .71502 .32167 L .7107 .32127 L .71019 .32122 L .70535 .32068 L .70052 .32006 L .69569 .31935 L .6926 .31885 L .69086 .31856 L .68603 .31767 L .68119 .31668 L .68008 .31644 L .67636 .31559 L .67153 .3144 L .67008 .31402 L .6667 .3131 L .66187 .31167 L .66164 .31161 L .65704 .31013 L .65431 .30919 L Mistroke .6522 .30844 L .64781 .30677 L .64737 .3066 L .64254 .3046 L .64198 .30436 L .63771 .30242 L .63671 .30194 L .63288 .30003 L .63192 .29953 L .62804 .2974 L .62754 .29711 L .62354 .29469 L .62321 .29449 L .61986 .29228 L .61838 .29124 L .61649 .28986 L .61355 .28757 L .6134 .28745 L .61057 .28503 L .60872 .28333 L .60798 .28261 L .60562 .2802 L .60388 .27826 L .60348 .27778 L .60154 .27537 L .5998 .27295 L .59905 .27182 L .59826 .27054 L .5969 .26812 L .59572 .2657 L .59471 .26329 L .59422 .26192 L .59388 .26087 L .59322 .25846 L .59273 .25604 L .5924 .25362 L .59223 .25121 L .59223 .24879 L .5924 .24638 L .59273 .24396 L .59322 .24154 L .59388 .23913 L .59422 .23808 L .59471 .23671 L .59572 .2343 L .5969 .23188 L .59826 .22946 L .59905 .22818 L .5998 .22705 L .60154 .22463 L Mistroke .60348 .22222 L .60388 .22174 L .60562 .2198 L .60798 .21739 L .60872 .21667 L .61057 .21497 L .6134 .21255 L .61355 .21243 L .61649 .21014 L .61838 .20876 L .61986 .20772 L .62321 .20551 L .62354 .20531 L .62754 .20289 L .62804 .2026 L .63192 .20047 L .63288 .19997 L .63671 .19806 L .63771 .19758 L .64198 .19564 L .64254 .1954 L .64737 .1934 L .64781 .19323 L .6522 .19156 L .65431 .19081 L .65704 .18987 L .66164 .18839 L .66187 .18833 L .6667 .1869 L .67008 .18598 L .67153 .1856 L .67636 .18441 L .68008 .18356 L .68119 .18332 L .68603 .18233 L .69086 .18144 L .6926 .18115 L .69569 .18065 L .70052 .17994 L .70535 .17932 L .71019 .17878 L .7107 .17873 L .71502 .17833 L Mfstroke P p .001 w .23183 .06976 m .23666 .06908 L .2415 .06853 L .24633 .06812 L .25116 .06783 L .25599 .06767 L .26082 .06763 L .26566 .06773 L .27049 .06795 L .27532 .06831 L .28015 .06879 L .28498 .0694 L .28899 .07001 L .28981 .07015 L .29465 .07103 L .29948 .07205 L .30115 .07243 L .30431 .0732 L .30914 .07449 L .31037 .07485 L .31397 .07593 L .31805 .07726 L .31881 .07752 L .32364 .07926 L .32474 .07968 L .32847 .08115 L .33072 .08209 L .3333 .08321 L .33616 .08451 L .33813 .08544 L .34116 .08693 L .34296 .08784 L .3458 .08934 L .3478 .09043 L .35014 .09176 L .35263 .09322 L .35421 .09417 L .35746 .09621 L .35805 .09659 L .36168 .099 L .36229 .09942 L .36513 .10142 L .36712 .10287 L .36841 .10384 L .37154 .10625 L .37196 .10659 L .37452 .10867 L .37679 .11058 L .37737 .11108 L .3801 .1135 L Mistroke .38162 .11489 L .38271 .11592 L .38521 .11833 L .38645 .11957 L .38761 .12075 L .38991 .12316 L .39128 .12466 L .39211 .12558 L .39422 .128 L .39612 .13026 L .39624 .13041 L .39817 .13283 L .40002 .13524 L .40095 .13649 L .40179 .13766 L .40348 .14008 L .40509 .14249 L .40578 .14356 L .40662 .14491 L .40808 .14732 L .40946 .14974 L .41061 .15186 L .41077 .15216 L .412 .15457 L .41316 .15699 L .41424 .1594 L .41526 .16182 L .41544 .16228 L .4162 .16423 L .41707 .16665 L .41786 .16907 L .41859 .17148 L .41924 .1739 L .41982 .17631 L .42027 .17842 L .42034 .17873 L .42078 .18115 L .42115 .18356 L .42146 .18598 L .42169 .18839 L .42186 .19081 L .42197 .19323 L .42201 .19564 L .422 .19806 L .42192 .20047 L .42179 .20289 L .42161 .20531 L .42138 .20772 L .42111 .21014 L .4208 .21255 L Mistroke .42046 .21497 L .42027 .21621 L .42009 .21739 L .4197 .2198 L .4193 .22222 L .41889 .22463 L .41849 .22705 L .4181 .22946 L .41772 .23188 L .41737 .2343 L .41706 .23671 L .41678 .23913 L .41655 .24154 L .41638 .24396 L .41626 .24638 L .4162 .24879 L .4162 .25121 L .41626 .25362 L .41638 .25604 L .41655 .25846 L .41678 .26087 L .41706 .26329 L .41737 .2657 L .41772 .26812 L .4181 .27054 L .41849 .27295 L .41889 .27537 L .4193 .27778 L .4197 .2802 L .42009 .28261 L .42027 .28379 L .42046 .28503 L .4208 .28745 L .42111 .28986 L .42138 .29228 L .42161 .29469 L .42179 .29711 L .42192 .29953 L .422 .30194 L .42201 .30436 L .42197 .30677 L .42186 .30919 L .42169 .31161 L .42146 .31402 L .42115 .31644 L .42078 .31885 L .42034 .32127 L .42027 .32158 L .41982 .32369 L .41924 .3261 L Mistroke .41859 .32852 L .41786 .33093 L .41707 .33335 L .4162 .33577 L .41544 .33772 L .41526 .33818 L .41424 .3406 L .41316 .34301 L .412 .34543 L .41077 .34784 L .41061 .34814 L .40946 .35026 L .40808 .35268 L .40662 .35509 L .40578 .35644 L .40509 .35751 L .40348 .35992 L .40179 .36234 L .40095 .36351 L .40002 .36476 L .39817 .36717 L .39624 .36959 L .39612 .36974 L .39422 .372 L .39211 .37442 L .39128 .37534 L .38991 .37684 L .38761 .37925 L .38645 .38043 L .38521 .38167 L .38271 .38408 L .38162 .38511 L .3801 .3865 L .37737 .38892 L .37679 .38942 L .37452 .39133 L .37196 .39341 L .37154 .39375 L .36841 .39616 L .36712 .39713 L .36513 .39858 L .36229 .40058 L .36168 .401 L .35805 .40341 L .35746 .40379 L .35421 .40583 L .35263 .40678 L .35014 .40824 L .3478 .40957 L .3458 .41066 L Mistroke .34296 .41216 L .34116 .41307 L .33813 .41456 L .33616 .41549 L .3333 .41679 L .33072 .41791 L .32847 .41885 L .32474 .42032 L .32364 .42074 L .31881 .42248 L .31805 .42274 L .31397 .42407 L .31037 .42515 L .30914 .42551 L .30431 .4268 L .30115 .42757 L .29948 .42795 L .29465 .42897 L .28981 .42985 L .28899 .42999 L .28498 .4306 L .28015 .43121 L .27532 .43169 L .27049 .43205 L .26566 .43227 L .26082 .43237 L .25599 .43233 L .25116 .43217 L .24633 .43188 L .2415 .43147 L .23666 .43092 L .23183 .43024 L .23024 .42999 L .227 .42943 L .22217 .42848 L .21808 .42757 L .21734 .4274 L .2125 .42617 L .20886 .42515 L .20767 .42481 L .20284 .42329 L .20118 .42274 L .19801 .42163 L .19449 .42032 L .19318 .41981 L .18851 .41791 L .18835 .41784 L .18351 .4157 L .18307 .41549 L .17868 .41338 L Mistroke .17807 .41307 L .17385 .41089 L .17343 .41066 L .16909 .40824 L .16902 .4082 L .16502 .40583 L .16419 .40531 L .16118 .40341 L .15935 .40221 L .15755 .401 L .15452 .39888 L .1541 .39858 L .15082 .39616 L .14969 .3953 L .14769 .39375 L .14486 .39145 L .14471 .39133 L .14186 .38892 L .14003 .3873 L .13913 .3865 L .13652 .38408 L .1352 .38282 L .13402 .38167 L .13162 .37925 L .13036 .37794 L .12933 .37684 L .12712 .37442 L .12553 .37261 L .12501 .372 L .12299 .36959 L .12106 .36717 L .1207 .36671 L .11921 .36476 L .11744 .36234 L .11587 .36009 L .11575 .35992 L .11414 .35751 L .11261 .35509 L .11116 .35268 L .11104 .35247 L .10978 .35026 L .10847 .34784 L .10724 .34543 L .1062 .34327 L .10608 .34301 L .105 .3406 L .10399 .33818 L .10305 .33577 L .10219 .33335 L .10139 .33093 L Mistroke .10137 .33087 L .10067 .32852 L .10002 .3261 L .09944 .32369 L .09894 .32127 L .0985 .31885 L .09813 .31644 L .09783 .31402 L .0976 .31161 L .09743 .30919 L .09733 .30677 L .09729 .30436 L .09732 .30194 L .0974 .29953 L .09753 .29711 L .09772 .29469 L .09795 .29228 L .09823 .28986 L .09855 .28745 L .09889 .28503 L .09926 .28261 L .09966 .2802 L .10006 .27778 L .10047 .27537 L .10088 .27295 L .10127 .27054 L .10137 .26991 L .10165 .26812 L .102 .2657 L .10232 .26329 L .10259 .26087 L .10282 .25846 L .103 .25604 L .10312 .25362 L .10318 .25121 L .10318 .24879 L .10312 .24638 L .103 .24396 L .10282 .24154 L .10259 .23913 L .10232 .23671 L .102 .2343 L .10165 .23188 L .10137 .23009 L .10127 .22946 L .10088 .22705 L .10047 .22463 L .10006 .22222 L .09966 .2198 L .09926 .21739 L Mistroke .09889 .21497 L .09855 .21255 L .09823 .21014 L .09795 .20772 L .09772 .20531 L .09753 .20289 L .0974 .20047 L .09732 .19806 L .09729 .19564 L .09733 .19323 L .09743 .19081 L .0976 .18839 L .09783 .18598 L .09813 .18356 L .0985 .18115 L .09894 .17873 L .09944 .17631 L .10002 .1739 L .10067 .17148 L .10137 .16913 L .10139 .16907 L .10219 .16665 L .10305 .16423 L .10399 .16182 L .105 .1594 L .10608 .15699 L .1062 .15673 L .10724 .15457 L .10847 .15216 L .10978 .14974 L .11104 .14753 L .11116 .14732 L .11261 .14491 L .11414 .14249 L .11575 .14008 L .11587 .13991 L .11744 .13766 L .11921 .13524 L .1207 .13329 L .12106 .13283 L .12299 .13041 L .12501 .128 L .12553 .12739 L .12712 .12558 L .12933 .12316 L .13036 .12206 L .13162 .12075 L .13402 .11833 L .1352 .11718 L .13652 .11592 L Mistroke .13913 .1135 L .14003 .1127 L .14186 .11108 L .14471 .10867 L .14486 .10855 L .14769 .10625 L .14969 .1047 L .15082 .10384 L .1541 .10142 L .15452 .10112 L .15755 .099 L .15935 .09779 L .16118 .09659 L .16419 .09469 L .16502 .09417 L .16902 .0918 L .16909 .09176 L .17343 .08934 L .17385 .08911 L .17807 .08693 L .17868 .08662 L .18307 .08451 L .18351 .0843 L .18835 .08216 L .18851 .08209 L .19318 .08019 L .19449 .07968 L .19801 .07837 L .20118 .07726 L .20284 .07671 L .20767 .07519 L .20886 .07485 L .2125 .07383 L .21734 .0726 L .21808 .07243 L .22217 .07152 L .227 .07057 L .23024 .07001 L .23183 .06976 L Mfstroke P P MathPictureEnd end %%EndDocument endTexFig 143 2499 a Fr(Figure)g(3:)22 b(Probabilit)o(y)15 b(densities)h(after)g (encoun)o(tering)f(a)i(T)o(yp)q(e)f(3)h(Av)o(oided)e(Crossing.)963 2753 y(7)p eop %%Page: 8 8 8 7 bop 0 -22 a Fr(gap)17 b(predicts)f(an)h(electronic)e(transition)i (probabilit)o(y)f(of)h Fn(e)1117 -40 y Ff(\000)p Fj(f)t Fm(\()p Fj(y)1196 -35 y Fd(2)1213 -40 y Fm(\))1245 -22 y Fr(as)h(the)e(system)f(mo)o(v)o(es)g(through)i(the)0 39 y(a)o(v)o(oided)e(crossing.)22 b(Theorem)15 b(3.1)i(con\014rms)e (this)h(in)o(tuition.)73 159 y(T)o(yp)q(e)d(4)g(Av)o(oided)f(Crossings) i(are)f(similar,)e(except)h(that)h(the)g(minimal)d(m)o(ultipli)o(ci)o (t)o(y)g(of)j(eigen)o(v)m(alues)0 219 y(allo)o(w)o(ed)h(b)o(y)h(the)g (symmetry)c(group)16 b(is)f(2.)21 b(Near)15 b(one)g(of)h(these)f(a)o(v) o(oided)f(crossings,)h(one)h(can)f(c)o(ho)q(ose)h(an)0 279 y(orthonormal)f(basis)h Fk(f)p Fn( )454 286 y Fm(1)474 279 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(;)g( )636 286 y Fm(2)654 279 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )816 286 y Fm(3)835 279 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )997 286 y Fm(4)1015 279 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fk(g)15 b Fr(of)h Fn(P)7 b Fr(\()p Fn(x;)h(\017)p Fr(\))p Fk(H)p Fr(,)15 b(whic)o(h)g(is)g(regular)h(in)f(\()p Fn(x;)8 b(\017)p Fr(\))0 339 y(around)17 b(\(0)p Fn(;)8 b Fr(0\).)22 b(The)16 b(op)q(erator)i Fn(h)634 347 y Ff(k)670 339 y Fr(satis\014es)122 447 y Fn(h)150 455 y Ff(k)170 447 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))13 b(=)h Fn(h)371 454 y Fm(1)390 447 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)566 431 y Fi(e)558 447 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))p Fn(;)1106 b Fr(\(1.14\))0 549 y(where)16 b Fn(h)169 556 y Fm(1)189 549 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))15 b(is)h(represen)o(ted)f(in)h(this)g(basis)h(b)o(y)f(the)g(matrix)130 612 y Fi(0)130 685 y(B)130 710 y(B)130 735 y(B)130 761 y(@)278 644 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))150 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))207 b(0)367 b(0)175 705 y Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))130 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))290 b(0)367 b(0)334 765 y(0)h(0)311 b Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))150 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))334 825 y(0)368 b(0)208 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))130 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))1701 612 y Fi(1)1701 685 y(C)1701 710 y(C)1701 735 y(C)1701 761 y(A)1746 735 y Fn(;)65 b Fr(\(1.15\))0 929 y(and)107 913 y Fi(e)99 929 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))20 b(is)g(represen)o(ted)f(b)o(y)659 910 y Fm(1)p 659 918 18 2 v 659 946 a(4)690 929 y Fr(trace)o(\()p Fn(h)p Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(P)f Fr(\()p Fn(x;)h(\017)p Fr(\)\))19 b(times)g(the)h(4)14 b Fk(\002)f Fr(4)21 b(iden)o(tit)o(y)d(matrix.)32 b(The)0 997 y(function)199 981 y Fi(e)191 997 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))16 b(is)g(regular)g(in)g(\()p Fn(x;)8 b(\017)p Fr(\))15 b(near)i(the)f(origin,)g(and)131 1098 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(b)410 1105 y Fm(1)430 1098 y Fn(x)458 1105 y Fm(1)488 1098 y Fr(+)11 b Fn(b)558 1105 y Fm(2)578 1098 y Fn(x)606 1105 y Fm(2)636 1098 y Fr(+)g Fn(b)706 1105 y Fm(3)726 1098 y Fn(\017)g Fr(+)g Fk(O)q Fr(\(2\))916 b(\(1.16\))133 1171 y Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(c)410 1178 y Fm(2)430 1171 y Fn(x)458 1178 y Fm(2)489 1171 y Fr(+)11 b Fn(c)559 1178 y Fm(3)578 1171 y Fn(\017)g Fr(+)g Fk(O)q Fr(\(2\))138 1244 y Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))40 b(=)h Fn(d)414 1251 y Fm(3)434 1244 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\(2\))130 1300 y Fi(e)122 1316 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 1418 y Fr(The)16 b(t)o(w)o(o)g(energy)g(lev)o(els)f(in)o(v)o (olv)o(ed)f(in)h(the)h(Av)o(oided)f(Crossing)j(are)e(th)o(us)122 1529 y Fn(E)5 b Fe(A)165 1557 y(B)236 1529 y Fr(=)324 1513 y Fi(e)316 1529 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 1477 y Fi(q)p 581 1477 747 2 v 581 1529 a Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))720 1515 y Fm(2)749 1529 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))934 1515 y Fm(2)964 1529 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1145 1515 y Fm(2)1175 1529 y Fr(+)j Fk(O)q Fr(\(2\))236 1630 y(=)324 1614 y Fi(e)316 1630 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 1577 y Fi(q)p 581 1577 1103 2 v 581 1630 a Fr(\()p Fn(b)621 1637 y Fm(1)641 1630 y Fn(x)669 1637 y Fm(1)699 1630 y Fr(+)11 b Fn(b)769 1637 y Fm(2)789 1630 y Fn(x)817 1637 y Fm(2)847 1630 y Fr(+)g Fn(b)917 1637 y Fm(3)937 1630 y Fn(\017)p Fr(\))976 1615 y Fm(2)1006 1630 y Fr(+)g(\()p Fn(c)1095 1637 y Fm(2)1115 1630 y Fn(x)1143 1637 y Fm(2)1173 1630 y Fr(+)g Fn(c)1243 1637 y Fm(3)1263 1630 y Fn(\017)p Fr(\))1302 1615 y Fm(2)1332 1630 y Fr(+)g(\()p Fn(d)1425 1637 y Fm(3)1445 1630 y Fn(\017)p Fr(\))1484 1615 y Fm(2)1523 1630 y Fr(+)19 b Fk(O)q Fr(\(3\))q Fn(:)127 b Fr(\(1.17\))73 1731 y(T)o(yp)q(e)21 b(5)h(Av)o(oided)e(Crossings)j(ha)o(v)o(e)e(the)g (co)q(dimension)f(of)h(\000)h(equal)f(to)h(3)g(and)g(the)f(m)o(ultipli) o(cit)n(y)0 1792 y(of)e(the)g(eigen)o(v)m(alues)g(equal)f(to)i(2.)30 b(W)l(e)19 b(c)o(ho)q(ose)h(an)f(orthogonal)i(co)q(ordinate)f(system)e (for)h(the)g(n)o(uclear)0 1852 y(con\014gurations)h(in)f(whic)o(h)f (the)h Fn(x)634 1859 y Fm(1)653 1852 y Fr(,)h Fn(x)715 1859 y Fm(2)734 1852 y Fr(,)f(and)h Fn(x)893 1859 y Fm(3)931 1852 y Fr(co)q(ordinate)g(directions)e(are)h(p)q(erp)q(endicular)g(to)g (\000)h(at)0 1912 y Fn(x)f Fr(=)f(0;)j(the)e Fn(x)277 1919 y Fm(4)297 1912 y Fr(,)g Fn(x)358 1919 y Fm(5)377 1912 y Fr(,)h Fn(:)8 b(:)g(:)g(x)505 1919 y Fj(n)547 1912 y Fr(co)q(ordinate)20 b(directions)e(are)i(parallel)e(to)i(\000)f (at)h Fn(x)e Fr(=)h(0;)i(and)f(so)g(that)f(the)0 1972 y(v)o(ector)c Fn(\021)172 1954 y Fm(0)208 1972 y Fr(has)i(the)f(form) 122 2255 y Fn(\021)148 2234 y Fm(0)181 2255 y Fr(=)233 2033 y Fi(0)233 2106 y(B)233 2131 y(B)233 2155 y(B)233 2180 y(B)233 2205 y(B)233 2230 y(B)233 2255 y(B)233 2280 y(B)233 2305 y(B)233 2330 y(B)233 2355 y(B)233 2381 y(@)279 2065 y Fn(\021)305 2047 y Fm(0)303 2077 y(1)289 2125 y Fr(0)289 2185 y(0)279 2245 y Fn(\021)305 2227 y Fm(0)303 2258 y(4)279 2306 y Fn(\021)305 2288 y Fm(0)303 2318 y(5)295 2352 y Fr(.)295 2368 y(.)295 2385 y(.)278 2445 y Fn(\021)304 2427 y Fm(0)302 2458 y Fj(n)334 2033 y Fi(1)334 2106 y(C)334 2131 y(C)334 2155 y(C)334 2180 y(C)334 2205 y(C)334 2230 y(C)334 2255 y(C)334 2280 y(C)334 2305 y(C)334 2330 y(C)334 2355 y(C)334 2381 y(A)378 2255 y Fn(:)1433 b Fr(\(1.18\))0 2541 y(W)l(e)15 b(can)h(c)o(ho)q(ose)g(an)f (orthonormal)h(basis)g Fk(f)p Fn( )847 2548 y Fm(1)866 2541 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(;)g( )1028 2548 y Fm(2)1046 2541 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )1208 2548 y Fm(3)1227 2541 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )1389 2548 y Fm(4)1407 2541 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fk(g)15 b Fr(of)h Fn(P)7 b Fr(\()p Fn(x;)h(\017)p Fr(\))p Fk(H)p Fr(,)15 b(whic)o(h)0 2601 y(is)23 b(regular)f(in)h(\()p Fn(x;)8 b(\017)p Fr(\))22 b(around)i(\(0)p Fn(;)8 b Fr(0\).)41 b(The)23 b(op)q(erator)h Fn(h)1094 2609 y Ff(k)1113 2601 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))22 b(satis\014es)h(\(1.14\),)i(where)d Fn(h)1767 2608 y Fm(1)1787 2601 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))22 b(is)963 2753 y(8)p eop %%Page: 9 9 9 8 bop 0 -22 a Fr(represen)o(ted)15 b(in)h(this)g(basis)h(b)o(y)f(the) g(matrix)130 39 y Fi(0)130 112 y(B)130 136 y(B)130 161 y(B)130 188 y(@)278 71 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))328 b(0)244 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))66 b Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))334 131 y(0)330 b Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))168 b Fk(\000)p Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))49 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))175 191 y Fn(\015)s Fr(\()p Fn(x;)g(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))47 b Fk(\000)p Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))149 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))308 b(0)176 252 y Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))j Fk(\000)f Fn(i\030)r Fr(\()p Fn(x;)e(\017)p Fr(\))68 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))242 b(0)310 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))1774 39 y Fi(1)1774 112 y(C)1774 136 y(C)1774 161 y(C)1774 188 y(A)1819 161 y Fn(;)1825 318 y Fr(\(1.19\))0 420 y(and)107 404 y Fi(e)99 420 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))20 b(is)g(represen)o(ted)f(b) o(y)659 400 y Fm(1)p 659 408 18 2 v 659 437 a(4)690 420 y Fr(trace)o(\()p Fn(h)p Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(P)f Fr(\()p Fn(x;)h(\017)p Fr(\)\))19 b(times)g(the)h(4)14 b Fk(\002)f Fr(4)21 b(iden)o(tit)o(y)d(matrix.)32 b(The)0 487 y(function)199 471 y Fi(e)191 487 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))16 b(is)g(regular)g(in)g(\()p Fn(x;)8 b(\017)p Fr(\))15 b(near)i(the)f(origin,)g(and)131 589 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(b)410 596 y Fm(1)430 589 y Fn(x)458 596 y Fm(1)488 589 y Fr(+)11 b Fn(b)558 596 y Fm(2)578 589 y Fn(x)606 596 y Fm(2)636 589 y Fr(+)g Fn(b)706 596 y Fm(3)726 589 y Fn(x)754 596 y Fm(3)784 589 y Fr(+)g Fn(b)854 596 y Fm(4)874 589 y Fn(\017)f Fr(+)h Fk(O)q Fr(\(2\))133 662 y Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(c)410 669 y Fm(2)430 662 y Fn(x)458 669 y Fm(2)489 662 y Fr(+)11 b Fn(c)559 669 y Fm(4)578 662 y Fn(\017)g Fr(+)g Fk(O)q Fr(\(2\))138 734 y Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))40 b(=)h Fn(d)414 741 y Fm(3)434 734 y Fn(x)462 741 y Fm(3)493 734 y Fr(+)11 b Fn(d)567 741 y Fm(4)587 734 y Fn(\017)g Fr(+)g Fk(O)q Fr(\(2\))136 807 y Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(e)412 814 y Fm(4)432 807 y Fn(\017)10 b Fr(+)h Fk(O)q Fr(\(2\))138 880 y Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(2\))130 936 y Fi(e)122 952 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 1054 y Fr(W)l(e)16 b(assume)g(the)h (generically)e(satis\014ed)i(condition)f Fn(e)1006 1061 y Fm(4)1040 1054 y Fk(6)p Fr(=)e(0.)23 b(The)17 b(t)o(w)o(o)g(energy)f (lev)o(els)e(in)o(v)o(olv)o(ed)h(in)h(the)0 1114 y(Av)o(oided)f (Crossing)i(are)g(th)o(us)243 1216 y Fn(E)5 b Fe(A)286 1244 y(B)163 1316 y Fr(=)251 1300 y Fi(e)243 1316 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)450 1264 y Fi(q)p 492 1264 1006 2 v 52 x Fn(\014)s Fr(\()p Fn(x;)e(\017)p Fr(\))631 1302 y Fm(2)660 1316 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))845 1302 y Fm(2)875 1316 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1056 1302 y Fm(2)1086 1316 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1268 1302 y Fm(2)1298 1316 y Fr(+)j Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))1478 1302 y Fm(2)163 1463 y Fr(=)251 1447 y Fi(e)243 1463 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)450 1357 y Fi(v)450 1380 y(u)450 1405 y(u)450 1430 y(u)450 1455 y(t)p 494 1357 1397 2 v -77 x(0)494 1452 y(@)552 1409 y Fm(3)531 1422 y Fi(X)530 1513 y Fj(j)r Fm(=1)600 1463 y Fn(b)621 1470 y Fj(j)639 1463 y Fn(x)667 1470 y Fj(j)704 1463 y Fr(+)h Fn(b)774 1470 y Fm(4)794 1463 y Fn(\017)814 1378 y Fi(1)814 1452 y(A)850 1389 y Fm(2)881 1463 y Fr(+)g(\()p Fn(c)970 1470 y Fm(2)989 1463 y Fn(x)1017 1470 y Fm(2)1048 1463 y Fr(+)g Fn(c)1118 1470 y Fm(4)1137 1463 y Fn(\017)p Fr(\))1176 1449 y Fm(2)1207 1463 y Fr(+)g(\()p Fn(d)1300 1470 y Fm(3)1320 1463 y Fn(x)1348 1470 y Fm(3)1378 1463 y Fr(+)g Fn(d)1452 1470 y Fm(4)1473 1463 y Fn(\017)p Fr(\))1512 1449 y Fm(2)1542 1463 y Fr(+)g(\()p Fn(e)1633 1470 y Fm(4)1652 1463 y Fn(\017)p Fr(\))1691 1449 y Fm(2)1730 1463 y Fr(+)19 b Fk(O)q Fr(\(3\))q Fn(:)1825 1583 y Fr(\(1.20\))73 1685 y(T)o(yp)q(e)i(6)h(Av)o(oided)e(Crossings)j(ha)o(v)o(e)e(the)g(co) q(dimension)f(of)h(\000)h(equal)f(to)h(4)g(and)g(the)f(m)o(ultipli)o (cit)n(y)0 1745 y(of)e(the)g(eigen)o(v)m(alues)g(equal)f(to)i(2.)30 b(W)l(e)19 b(c)o(ho)q(ose)h(an)f(orthogonal)i(co)q(ordinate)f(system)e (for)h(the)g(n)o(uclear)0 1806 y(con\014gurations)g(in)f(whic)o(h)f (the)g Fn(x)629 1813 y Fm(1)648 1806 y Fr(,)h Fn(x)708 1813 y Fm(2)728 1806 y Fr(,)f Fn(x)787 1813 y Fm(3)807 1806 y Fr(,)g(and)i Fn(x)963 1813 y Fm(4)1000 1806 y Fr(co)q(ordinate)f(directions)f(are)h(p)q(erp)q(endicular)f(to)i(\000)0 1866 y(at)e Fn(x)e Fr(=)f(0;)j(the)g Fn(x)323 1873 y Fm(5)342 1866 y Fr(,)f Fn(x)400 1873 y Fm(6)420 1866 y Fr(,)g Fn(:)8 b(:)g(:)g(x)544 1873 y Fj(n)584 1866 y Fr(co)q(ordinate)17 b(directions)f(are)h(parallel)f(to)h(\000)g(at)g Fn(x)e Fr(=)f(0;)j(and)h(so)f(that)g(the)0 1926 y(v)o(ector)e Fn(\021)172 1908 y Fm(0)208 1926 y Fr(has)i(the)f(form)122 2239 y Fn(\021)148 2218 y Fm(0)181 2239 y Fr(=)233 1991 y Fi(0)233 2065 y(B)233 2089 y(B)233 2114 y(B)233 2139 y(B)233 2164 y(B)233 2189 y(B)233 2214 y(B)233 2239 y(B)233 2264 y(B)233 2289 y(B)233 2314 y(B)233 2339 y(B)233 2363 y(B)233 2390 y(@)279 2019 y Fn(\021)305 2001 y Fm(0)303 2031 y(1)289 2079 y Fr(0)289 2139 y(0)289 2199 y(0)279 2259 y Fn(\021)305 2241 y Fm(0)303 2272 y(5)279 2320 y Fn(\021)305 2301 y Fm(0)303 2332 y(6)295 2366 y Fr(.)295 2382 y(.)295 2399 y(.)278 2459 y Fn(\021)304 2441 y Fm(0)302 2471 y Fj(n)334 1991 y Fi(1)334 2065 y(C)334 2089 y(C)334 2114 y(C)334 2139 y(C)334 2164 y(C)334 2189 y(C)334 2214 y(C)334 2239 y(C)334 2264 y(C)334 2289 y(C)334 2314 y(C)334 2339 y(C)334 2363 y(C)334 2390 y(A)378 2239 y Fn(:)1433 b Fr(\(1.21\))0 2555 y(W)l(e)15 b(can)h(c)o(ho)q(ose)g(an)f (orthonormal)h(basis)g Fk(f)p Fn( )847 2562 y Fm(1)866 2555 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(;)g( )1028 2562 y Fm(2)1046 2555 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )1208 2562 y Fm(3)1227 2555 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)g( )1389 2562 y Fm(4)1407 2555 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fk(g)15 b Fr(of)h Fn(P)7 b Fr(\()p Fn(x;)h(\017)p Fr(\))p Fk(H)p Fr(,)15 b(whic)o(h)0 2615 y(is)23 b(regular)f(in)h(\()p Fn(x;)8 b(\017)p Fr(\))22 b(around)i(\(0)p Fn(;)8 b Fr(0\).)41 b(The)23 b(op)q(erator)h Fn(h)1094 2623 y Ff(k)1113 2615 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))22 b(satis\014es)h(\(1.14\),)i(where)d Fn(h)1767 2622 y Fm(1)1787 2615 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))22 b(is)963 2753 y(9)p eop %%Page: 10 10 10 9 bop 0 -22 a Fr(represen)o(ted)15 b(in)h(this)g(basis)h(b)o(y)f (the)g(matrix)130 39 y Fi(0)130 112 y(B)130 136 y(B)130 161 y(B)130 188 y(@)278 71 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))328 b(0)244 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))66 b Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))334 131 y(0)330 b Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))168 b Fk(\000)p Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))49 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))175 191 y Fn(\015)s Fr(\()p Fn(x;)g(\017)p Fr(\))i Fk(\000)h Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))47 b Fk(\000)p Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))i Fk(\000)h Fn(i\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))149 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))308 b(0)176 252 y Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))j Fk(\000)f Fn(i\030)r Fr(\()p Fn(x;)e(\017)p Fr(\))68 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))j(+)g Fn(i\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))242 b(0)310 b Fk(\000)p Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))1774 39 y Fi(1)1774 112 y(C)1774 136 y(C)1774 161 y(C)1774 188 y(A)1819 161 y Fn(;)1825 318 y Fr(\(1.22\))0 420 y(and)107 404 y Fi(e)99 420 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))20 b(is)g(represen)o(ted)f(b) o(y)659 400 y Fm(1)p 659 408 18 2 v 659 437 a(4)690 420 y Fr(trace)o(\()p Fn(h)p Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(P)f Fr(\()p Fn(x;)h(\017)p Fr(\)\))19 b(times)g(the)h(4)14 b Fk(\002)f Fr(4)21 b(iden)o(tit)o(y)d(matrix.)32 b(The)0 487 y(function)199 471 y Fi(e)191 487 y Fn(V)11 b 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Fm(4)490 807 y Fr(+)11 b Fn(e)562 814 y Fm(5)582 807 y Fn(\017)f Fr(+)h Fk(O)q Fr(\(2\))138 880 y Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(f)413 887 y Fm(5)433 880 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\(2\))130 936 y Fi(e)122 952 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 1054 y Fr(W)l(e)16 b(assume)g(the)h(generically)d(satis\014ed)j (condition)g Fn(f)1007 1061 y Fm(5)1041 1054 y Fk(6)p Fr(=)d(0.)23 b(The)16 b(t)o(w)o(o)h(energy)f(lev)o(els)f(in)o(v)o(olv)o (ed)f(in)i(the)0 1114 y(Av)o(oided)f(Crossing)i(are)g(th)o(us)101 1216 y Fn(E)5 b Fe(A)144 1244 y(B)22 1316 y Fr(=)109 1300 y Fi(e)101 1316 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))j Fk(\006)309 1264 y Fi(q)p 350 1264 1006 2 v 350 1316 a Fn(\014)s Fr(\()p Fn(x;)d(\017)p Fr(\))489 1302 y Fm(2)518 1316 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))703 1302 y Fm(2)733 1316 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))914 1302 y Fm(2)944 1316 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1126 1302 y Fm(2)1156 1316 y Fr(+)j Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))1336 1302 y Fm(2)22 1400 y Fr(=)109 1383 y Fi(e)101 1400 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))101 1538 y Fk(\006)164 1432 y Fi(v)164 1455 y(u)164 1480 y(u)164 1505 y(u)164 1530 y(t)p 208 1432 1703 2 v -77 x(0)208 1528 y(@)266 1484 y Fm(4)245 1497 y Fi(X)245 1588 y Fj(j)r Fm(=1)314 1538 y Fn(b)335 1545 y Fj(j)353 1538 y Fn(x)381 1545 y Fj(j)410 1538 y Fr(+)j Fn(b)480 1545 y Fm(5)500 1538 y Fn(\017)520 1453 y Fi(1)520 1528 y(A)556 1464 y Fm(2)586 1538 y Fr(+)g(\()p Fn(c)675 1545 y Fm(2)695 1538 y Fn(x)723 1545 y Fm(2)754 1538 y Fr(+)g Fn(c)824 1545 y Fm(5)843 1538 y Fn(\017)p Fr(\))882 1524 y Fm(2)913 1538 y Fr(+)g(\()p Fn(d)1006 1545 y Fm(3)1026 1538 y Fn(x)1054 1545 y Fm(3)1084 1538 y Fr(+)g Fn(d)1158 1545 y Fm(5)1178 1538 y Fn(\017)p Fr(\))1217 1524 y Fm(2)1248 1538 y Fr(+)g(\()p Fn(e)1339 1545 y Fm(4)1358 1538 y Fn(x)1386 1545 y Fm(4)1417 1538 y Fr(+)g Fn(e)1489 1545 y Fm(5)1508 1538 y Fn(\017)p Fr(\))1547 1524 y Fm(2)1577 1538 y Fr(+)g(\()p Fn(f)1669 1545 y Fm(5)1689 1538 y Fn(\017)p Fr(\))1728 1524 y Fm(2)1759 1538 y Fr(+)g Fk(O)q Fr(\(3\))p Fn(:)1825 1659 y Fr(\(1.23\))73 1790 y(The)k(pap)q(er)h(is)f(organized)g(as)h(follo)o(ws:)21 b(In)15 b(Section)f(2)i(w)o(e)e(discuss)i(the)f(ordinary)g(di\013eren)o (tial)f(equa-)0 1851 y(tions)22 b(whose)h(solutions)f(will)f(b)q(e)h (used)g(to)g(describ)q(e)f(the)h(semiclassical)d(motion)i(of)h(the)g(n) o(uclei.)36 b(In)0 1911 y(Section)20 b(3)h(w)o(e)f(discuss)g (semiclassical)e(n)o(uclear)h(w)o(a)o(v)o(e)h(pac)o(k)o(ets)f(and)i (adiabatic)g(motion)e(of)i(the)f(elec-)0 1971 y(trons.)34 b(W)l(e)21 b(then)f(state)h(and)g(pro)o(v)o(e)e(our)i(main)e(result)h (for)h(T)o(yp)q(e)f(3)h(Av)o(oided)e(Crossings,)j(Theorem)0 2031 y(3.1.)f(In)15 b(Section)g(4)h(w)o(e)f(state)h(our)g(main)e (results,)h(Theorems)g(4.1,)g(4.2,)h(and)g(4.3,)f(for)h(T)o(yp)q(e)f (4,)h(5,)g(and)g(6)0 2091 y(Av)o(oided)f(Crossings,)i(resp)q(ectiv)o (ely)l(.)0 2241 y Fg(Ac)n(kno)n(wledgemen)n(ts.)73 2362 y Fr(George)d(Hagedorn)g(wishes)f(to)g(thank)h(the)f(Cen)o(tre)f(de)h (Ph)o(ysique)g(Th)o(\023)-23 b(eorique,)12 b(C.N.R.S.,)g(Marseille)0 2422 y(for)22 b(its)f(hospitalit)o(y)g(during)h(July)f(1994)j(and)e(Ma) o(y)f(1996.)39 b(Alain)21 b(Jo)o(y)o(e)g(wishes)h(to)g(thank)g (Virginia)0 2482 y(P)o(olytec)o(hnic)17 b(Institute)h(and)i(State)g (Univ)o(ersit)o(y)c(for)k(its)f(hospitalit)o(y)f(during)h(August{Octob) q(er)h(1994)0 2542 y(and)g(July{Septem)o(b)q(er)d(1995,)k(and)f(the)f (F)l(onds)h(National)g(Suisse)f(de)g(la)g(Rec)o(herc)o(he)e(Scien)o (ti\014que)h(for)0 2602 y(\014nancial)e(supp)q(ort.)951 2753 y(10)p eop %%Page: 11 11 11 10 bop 0 -22 a Fc(1.1)70 b(Con)n(v)n(enien)n(t)21 b(Changes)j(of)f(V)-6 b(ariables)0 71 y Fr(W)l(e)18 b(b)q(egin)h(with)g (T)o(yp)q(e)f(3)h(Av)o(oided)e(Crossings.)30 b(It)18 b(is)h(con)o(v)o(enien)o(t)d(to)j(remo)o(v)o(e)d(the)j Fn(\017)p Fr(-dep)q(endence)e(in)0 131 y(the)k(leading)h(order)g(of)f Fn(\014)s Fr(\()p 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y(\001)p Fn( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))i(+)h Fn(h)p Fr(\()p Fn(x;)d(\017)p Fr(\))p Fn( )r Fr(\()p Fn(x;)g(t)p Fr(\))783 b(\(1.29\))0 1164 y(for)17 b Fn( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))14 b(b)q(ecomes)122 1295 y Fn(i\017)159 1274 y Fm(2)192 1261 y Fn(@)p 183 1283 47 2 v 183 1329 a(@)s(t)234 1295 y(\036)p Fr(\()p Fn(x)310 1274 y Ff(0)321 1295 y Fn(;)8 b(t)p Fr(\))13 b(=)h Fk(\000)489 1261 y Fn(\017)509 1243 y Fm(4)p 489 1283 40 2 v 496 1329 a Fr(2)533 1295 y(\001)p Fn(\036)p Fr(\()p Fn(x)650 1274 y Ff(0)661 1295 y Fn(;)8 b(t)p Fr(\))j(+)g Fn(h)p Fr(\()p Fn(x)p Fr(\()p Fn(x)902 1274 y Ff(0)913 1295 y Fn(;)d(\017)p Fr(\))p Fn(;)g(\017)p Fr(\))p Fn(\036)p Fr(\()p Fn(x)1111 1274 y Ff(0)1121 1295 y Fn(;)g(t)p Fr(\))p Fn(;)631 b Fr(\(1.30\))0 1409 y(for)122 1511 y Fn(\036)p Fr(\()p Fn(x)198 1490 y Ff(0)209 1511 y Fn(;)8 b(t)p Fr(\))13 b(=)h Fn( )r Fr(\()p Fn(x)p Fr(\()p Fn(x)461 1490 y Ff(0)472 1511 y Fn(;)8 b(\017)p Fr(\))p Fn(;)g(t)p Fr(\))p Fn(;)1219 b Fr(\(1.31\))0 1613 y(with)122 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Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(c)410 2162 y Fm(2)430 2155 y Fn(x)458 2162 y Fm(2)489 2155 y Fr(+)11 b Fk(O)q Fr(\(2\))138 2227 y Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))40 b(=)h Fn(d)414 2234 y Fm(3)434 2227 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\(2\))130 2284 y Fi(e)122 2300 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 2402 y Fr(In)16 b(the)g(new)g(v)m (ariables,)g(the)g(t)o(w)o(o)g(relev)m(an)o(t)f(energy)h(lev)o(els)e (are)122 2513 y Fn(E)5 b Fe(A)165 2541 y(B)236 2513 y Fr(=)324 2497 y Fi(e)316 2513 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 2460 y Fi(q)p 581 2460 583 2 v 581 2513 a Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))720 2499 y Fm(2)749 2513 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))934 2499 y Fm(2)964 2513 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1145 2499 y Fm(2)236 2613 y Fr(=)324 2597 y Fi(e)316 2613 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 2561 y Fi(q)p 581 2561 862 2 v 581 2613 a Fr(\()p Fn(b)621 2620 y Fm(1)641 2613 y Fn(x)669 2620 y Fm(1)699 2613 y Fr(+)11 b Fn(b)769 2620 y Fm(2)789 2613 y Fn(x)817 2620 y Fm(2)836 2613 y Fr(\))855 2599 y Fm(2)886 2613 y Fr(+)g(\()p Fn(c)975 2620 y Fm(2)994 2613 y Fn(x)1022 2620 y Fm(2)1042 2613 y Fr(\))1061 2599 y Fm(2)1092 2613 y Fr(+)g(\()p Fn(d)1185 2620 y Fm(3)1205 2613 y Fn(\017)p Fr(\))1244 2599 y Fm(2)1282 2613 y Fr(+)20 b Fk(O)q Fr(\(3\))p Fn(:)368 b Fr(\(1.34\))951 2753 y(11)p eop %%Page: 12 12 12 11 bop 73 -22 a Fr(F)l(or)15 b(T)o(yp)q(e)f(4)h(Av)o(oided)e (Crossings,)j(w)o(e)e(mak)o(e)e(the)i(same)g(c)o(hange)g(of)h(v)m (ariables,)f(whic)o(h)g(leads)g(to)h(the)0 39 y(matrix)i(\(1.15\))j (for)f Fn(h)410 46 y Fm(1)448 39 y Fr(with)g(leading)g(b)q(eha)o(vior)f (for)h Fn(\014)s Fr(,)g Fn(\015)s Fr(,)g Fn(\016)r Fr(,)f(and)1298 22 y Fi(e)1290 39 y Fn(V)30 b Fr(giv)o(en)18 b(b)o(y)g(\(1.33\).)30 b(In)18 b(the)h(new)0 99 y(v)m(ariables,)d(the)g(t)o(w)o(o)g(relev)m (an)o(t)f(energy)h(lev)o(els)e(are)122 210 y Fn(E)5 b Fe(A)165 238 y(B)236 210 y Fr(=)324 193 y Fi(e)316 210 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 157 y Fi(q)p 581 157 583 2 v 581 210 a Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))720 195 y Fm(2)749 210 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))934 195 y Fm(2)964 210 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1145 195 y Fm(2)236 310 y Fr(=)324 294 y Fi(e)316 310 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)540 258 y Fi(q)p 581 258 862 2 v 581 310 a Fr(\()p Fn(b)621 317 y Fm(1)641 310 y Fn(x)669 317 y Fm(1)699 310 y Fr(+)11 b Fn(b)769 317 y Fm(2)789 310 y Fn(x)817 317 y Fm(2)836 310 y Fr(\))855 296 y Fm(2)886 310 y Fr(+)g(\()p Fn(c)975 317 y Fm(2)994 310 y Fn(x)1022 317 y Fm(2)1042 310 y Fr(\))1061 296 y Fm(2)1092 310 y Fr(+)g(\()p Fn(d)1185 317 y Fm(3)1205 310 y Fn(\017)p Fr(\))1244 296 y Fm(2)1282 310 y Fr(+)20 b Fk(O)q Fr(\(3\))p Fn(:)368 b Fr(\(1.35\))73 411 y(F)l(or)15 b(T)o(yp)q(e)f(5)h(Av)o(oided)f(Crossings,)h(w)o(e)f(mak)o(e)f(the)h(c) o(hange)h(of)g(v)m(ariables)f(de\014ned)h(implici)o(tly)c(b)o(y)j(the)0 471 y(follo)o(wing:)122 573 y Fn(b)143 580 y Fm(1)162 573 y Fn(x)190 552 y Ff(0)190 585 y Fm(1)229 573 y Fr(+)20 b Fn(b)308 580 y Fm(2)327 573 y Fn(x)355 552 y Ff(0)355 585 y Fm(2)394 573 y Fr(+)f Fn(b)472 580 y Fm(3)492 573 y Fn(x)520 552 y Ff(0)520 585 y Fm(3)580 573 y Fr(=)42 b Fn(b)681 580 y Fm(1)700 573 y Fn(x)728 580 y Fm(1)767 573 y Fr(+)19 b Fn(b)845 580 y Fm(2)865 573 y Fn(x)893 580 y Fm(2)932 573 y Fr(+)g Fn(b)1010 580 y Fm(3)1030 573 y Fn(x)1058 580 y Fm(3)1096 573 y Fr(+)h Fn(b)1175 580 y Fm(4)1194 573 y Fn(\017)451 645 y(c)472 652 y Fm(2)492 645 y Fn(x)520 625 y Ff(0)520 658 y Fm(2)580 645 y Fr(=)42 b Fn(c)681 652 y Fm(2)701 645 y Fn(x)729 652 y Fm(2)767 645 y Fr(+)20 b Fn(c)846 652 y Fm(4)865 645 y Fn(\017)446 718 y(d)471 725 y Fm(3)492 718 y Fn(x)520 697 y Ff(0)520 730 y Fm(3)580 718 y Fr(=)42 b Fn(d)685 725 y Fm(3)705 718 y Fn(x)733 725 y Fm(3)772 718 y Fr(+)19 b Fn(d)854 725 y Fm(4)874 718 y Fn(\017:)917 b Fr(\(1.36\))0 819 y(This)21 b(c)o(hange)g(of)g(v)m(ariables)g(leads)g(to)g(the)g(Sc)o (hr\177)-24 b(odinger)21 b(equation)g(\(1.30\))g(with)g Fn(h)1594 826 y Fm(1)1635 819 y Fr(represen)o(ted)f(b)o(y)0 879 y(\(1.19\))d(with)f(the)g(follo)o(wing)g(lo)q(cal)g(b)q(eha)o(vior) g(around)i Fn(x)13 b Fr(=)h(0)j(and)f Fn(\017)e Fr(=)g(0:)131 981 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(b)410 988 y Fm(1)430 981 y Fn(x)458 988 y Fm(1)488 981 y Fr(+)11 b Fn(b)558 988 y Fm(2)578 981 y Fn(x)606 988 y Fm(2)636 981 y Fr(+)g Fn(b)706 988 y Fm(3)726 981 y Fn(x)754 988 y Fm(3)784 981 y Fr(+)g Fk(O)q Fr(\(2\))889 b(\(1.37\))133 1053 y Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(c)410 1060 y Fm(2)430 1053 y Fn(x)458 1060 y Fm(2)489 1053 y Fr(+)11 b Fk(O)q Fr(\(2\))138 1126 y Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))40 b(=)h Fn(d)414 1133 y Fm(3)434 1126 y Fn(x)462 1133 y Fm(3)493 1126 y Fr(+)11 b Fk(O)q Fr(\(2\))136 1199 y Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(e)412 1206 y Fm(4)432 1199 y Fn(\017)10 b Fr(+)h Fk(O)q Fr(\(2\))138 1271 y Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(2\))130 1328 y Fi(e)122 1344 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 1445 y Fr(In)16 b(the)g(new)g(v)m(ariables,)g(the)g(t)o(w)o(o)g (relev)m(an)o(t)f(energy)h(lev)o(els)e(are)243 1546 y Fn(E)5 b Fe(A)286 1575 y(B)163 1647 y Fr(=)251 1631 y Fi(e)243 1647 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)450 1594 y Fi(q)p 492 1594 1006 2 v 53 x Fn(\014)s Fr(\()p Fn(x;)e(\017)p Fr(\))631 1633 y Fm(2)660 1647 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))845 1633 y Fm(2)875 1647 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1056 1633 y Fm(2)1086 1647 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1268 1633 y Fm(2)1298 1647 y Fr(+)j Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))1478 1633 y Fm(2)163 1740 y Fr(=)251 1724 y Fi(e)243 1740 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)450 1687 y Fi(q)p 492 1687 1218 2 v 53 x Fr(\()p Fn(b)532 1747 y Fm(1)551 1740 y Fn(x)579 1747 y Fm(1)610 1740 y Fr(+)h Fn(b)680 1747 y Fm(2)699 1740 y Fn(x)727 1747 y Fm(2)758 1740 y Fr(+)g Fn(b)828 1747 y Fm(3)847 1740 y Fn(x)875 1747 y Fm(3)895 1740 y Fr(\))914 1726 y Fm(2)944 1740 y Fr(+)g(\()p Fn(c)1033 1747 y Fm(2)1053 1740 y Fn(x)1081 1747 y Fm(2)1100 1740 y Fr(\))1119 1726 y Fm(2)1150 1740 y Fr(+)g(\()p Fn(d)1243 1747 y Fm(3)1263 1740 y Fn(x)1291 1747 y Fm(3)1311 1740 y Fr(\))1330 1726 y Fm(2)1360 1740 y Fr(+)g(\()p Fn(e)1451 1747 y Fm(4)1471 1740 y Fn(\017)p Fr(\))1510 1726 y Fm(2)1548 1740 y Fr(+)20 b Fk(O)q Fr(\(3\))p Fn(:)1825 1813 y Fr(\(1.38\))73 1914 y(F)l(or)15 b(T)o(yp)q(e)f(6)h(Av)o(oided)f(Crossings,)h(w)o(e)f (mak)o(e)f(the)h(c)o(hange)h(of)g(v)m(ariables)f(de\014ned)h(implici)o (tly)c(b)o(y)j(the)0 1974 y(follo)o(wing:)122 2075 y Fn(b)143 2082 y Fm(1)162 2075 y Fn(x)190 2055 y Ff(0)190 2088 y Fm(1)229 2075 y Fr(+)20 b Fn(b)308 2082 y Fm(2)327 2075 y Fn(x)355 2055 y Ff(0)355 2088 y Fm(2)394 2075 y Fr(+)f Fn(b)472 2082 y Fm(3)492 2075 y Fn(x)520 2055 y Ff(0)520 2088 y Fm(3)558 2075 y Fr(+)h Fn(b)637 2082 y Fm(4)656 2075 y Fn(x)684 2055 y Ff(0)684 2088 y Fm(4)745 2075 y Fr(=)41 b Fn(b)845 2082 y Fm(1)865 2075 y Fn(x)893 2082 y Fm(1)932 2075 y Fr(+)19 b Fn(b)1010 2082 y Fm(2)1030 2075 y Fn(x)1058 2082 y Fm(2)1096 2075 y Fr(+)h Fn(b)1175 2082 y Fm(3)1194 2075 y Fn(x)1222 2082 y Fm(3)1261 2075 y Fr(+)f Fn(b)1339 2082 y Fm(4)1359 2075 y Fn(x)1387 2082 y Fm(4)1425 2075 y Fr(+)h Fn(b)1504 2082 y Fm(5)1523 2075 y Fn(\017)615 2148 y(c)636 2155 y Fm(2)656 2148 y Fn(x)684 2127 y Ff(0)684 2160 y Fm(2)745 2148 y Fr(=)41 b Fn(c)845 2155 y Fm(2)865 2148 y Fn(x)893 2155 y Fm(2)932 2148 y Fr(+)19 b Fn(c)1010 2155 y Fm(5)1030 2148 y Fn(\017)611 2221 y(d)636 2228 y Fm(3)656 2221 y Fn(x)684 2200 y Ff(0)684 2233 y Fm(3)745 2221 y Fr(=)41 b Fn(d)849 2228 y Fm(3)870 2221 y Fn(x)898 2228 y Fm(3)936 2221 y Fr(+)20 b Fn(d)1019 2228 y Fm(5)1039 2221 y Fn(\017)614 2293 y(e)637 2300 y Fm(4)656 2293 y Fn(x)684 2273 y Ff(0)684 2306 y Fm(4)745 2293 y Fr(=)41 b Fn(e)847 2300 y Fm(4)867 2293 y Fn(x)895 2300 y Fm(4)934 2293 y Fr(+)19 b Fn(e)1014 2300 y Fm(5)1033 2293 y Fn(\017:)758 b Fr(\(1.39\))0 2394 y(This)21 b(c)o(hange)g(of)g (v)m(ariables)g(leads)g(to)g(the)g(Sc)o(hr\177)-24 b(odinger)21 b(equation)g(\(1.30\))g(with)g Fn(h)1594 2401 y Fm(1)1635 2394 y Fr(represen)o(ted)f(b)o(y)0 2455 y(\(1.22\))d(with)f(the)g (follo)o(wing)g(lo)q(cal)g(b)q(eha)o(vior)g(around)i Fn(x)13 b Fr(=)h(0)j(and)f Fn(\017)e Fr(=)g(0:)131 2556 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(b)410 2563 y Fm(1)430 2556 y Fn(x)458 2563 y Fm(1)488 2556 y Fr(+)11 b Fn(b)558 2563 y Fm(2)578 2556 y Fn(x)606 2563 y Fm(2)636 2556 y Fr(+)g Fn(b)706 2563 y Fm(3)726 2556 y Fn(x)754 2563 y Fm(3)784 2556 y Fr(+)g Fn(b)854 2563 y Fm(4)874 2556 y Fn(x)902 2563 y Fm(4)932 2556 y Fr(+)g Fk(O)q Fr(\(2\))741 b(\(1.40\))133 2629 y Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(c)410 2636 y Fm(2)430 2629 y Fn(x)458 2636 y Fm(2)489 2629 y Fr(+)11 b Fk(O)q Fr(\(2\))951 2753 y(12)p eop %%Page: 13 13 13 12 bop 138 -22 a Fn(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(d)414 -15 y Fm(3)434 -22 y Fn(x)462 -15 y Fm(3)493 -22 y Fr(+)11 b Fk(O)q Fr(\(2\))136 51 y Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(e)412 58 y Fm(4)432 51 y Fn(x)460 58 y Fm(4)490 51 y Fr(+)11 b Fk(O)q Fr(\(2\))138 124 y Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fn(f)413 131 y Fm(5)433 124 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\(2\))130 180 y Fi(e)122 196 y Fn(V)g Fr(\()p Fn(x;)d(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\(0\))p Fn(:)0 282 y Fr(In)16 b(the)g(new)g(v)m(ariables,)g(the)g(t)o(w)o(o)g (relev)m(an)o(t)f(energy)h(lev)o(els)e(are)243 367 y Fn(E)5 b Fe(A)286 395 y(B)163 467 y Fr(=)251 451 y Fi(e)243 467 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)450 415 y Fi(q)p 492 415 1006 2 v 52 x Fn(\014)s Fr(\()p Fn(x;)e(\017)p Fr(\))631 453 y Fm(2)660 467 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))845 453 y Fm(2)875 467 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))1056 453 y Fm(2)1086 467 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1268 453 y Fm(2)1298 467 y Fr(+)j Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))1478 453 y Fm(2)163 614 y Fr(=)251 598 y Fi(e)243 614 y Fn(V)j Fr(\()p Fn(x;)d(\017)p Fr(\))19 b Fk(\006)467 508 y Fi(v)467 531 y(u)467 556 y(u)467 581 y(u)467 606 y(t)p 511 508 1232 2 v -77 x(0)511 603 y(@)569 560 y Fm(4)548 573 y Fi(X)547 664 y Fj(j)r Fm(=1)617 614 y Fn(b)638 621 y Fj(j)656 614 y Fn(x)684 621 y Fj(j)702 529 y Fi(1)702 603 y(A)738 540 y Fm(2)769 614 y Fr(+)11 b(\()p Fn(c)858 621 y Fm(2)877 614 y Fn(x)905 621 y Fm(2)925 614 y Fr(\))944 600 y Fm(2)975 614 y Fr(+)g(\()p Fn(d)1068 621 y Fm(3)1088 614 y Fn(x)1116 621 y Fm(3)1135 614 y Fr(\))1154 600 y Fm(2)1185 614 y Fr(+)g(\()p Fn(e)1276 621 y Fm(4)1295 614 y Fn(x)1323 621 y Fm(4)1342 614 y Fr(\))1361 600 y Fm(2)1392 614 y Fr(+)g(\()p Fn(f)1484 621 y Fm(5)1504 614 y Fn(\017)p Fr(\))1543 600 y Fm(2)1582 614 y Fr(+)19 b Fk(O)q Fr(\(3\))p Fn(:)69 b Fr(\(1.41\))0 807 y Fo(2)83 b(Ordinary)17 b(Di\013eren)n(tial)h(Equations)g(of)g (Semiclassical)e(Me-)124 898 y(c)n(hanics)0 1008 y Fr(In)21 b(Section)g(3.1)h(w)o(e)g(in)o(tro)q(duce)f(semiclassical)e(w)o(a)o(v)o (e)h(pac)o(k)o(ets)h(for)h(the)f(n)o(uclei.)36 b(The)22 b(leading)f(order)0 1068 y(semiclassical)11 b(motion)i(for)g(these)h(w) o(a)o(v)o(e)e(pac)o(k)o(ets)h(is)g(determined)e(b)o(y)i(the)g (solutions)h(to)g(certain)f(systems)0 1128 y(of)23 b(ordinary)g (di\013eren)o(tial)e(equations.)40 b(These)22 b(in)o(v)o(olv)o(e)f (classical)g(mec)o(hanics,)h(the)g(classical)g(action)0 1188 y(asso)q(ciated)15 b(with)f(a)g(classical)g(tra)s(jectory)l(,)f (and)h(the)g(dynamics)f(of)h(certain)f(matrices)f(that)j(describ)q(e)e (the)0 1248 y(p)q(osition)21 b(and)g(momen)o(tum)16 b(uncertain)o(ties) j(of)i(the)f(w)o(a)o(v)o(e)f(pac)o(k)o(ets.)32 b(The)21 b(goal)g(of)f(this)h(section)f(is)g(to)0 1309 y(study)15 b(the)g(small)f Fk(j)p Fn(t)p Fk(j)g Fr(and)i Fn(\017)f 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b(presen)o(t)f(the)h(detailed)e(analysis)j(for)f(T)o(yp)q(e) f(3)i(Av)o(oided)d(Crossings.)46 b(A)o(t)23 b(the)g(end)h(of)g(this)0 1790 y(section)13 b(w)o(e)h(describ)q(e)f(what)h(mo)q(di\014cations)f (need)g(to)h(b)q(e)g(made)f(to)h(handle)f(the)h(other)f(t)o(yp)q(es)h (of)g(a)o(v)o(oided)0 1850 y(crossings.)73 1910 y(W)l(e)i(de\014ne)122 1996 y Fn(V)166 1960 y Fe(A)168 1988 y(B)198 1996 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))13 b(=)379 1980 y Fi(e)371 1996 y Fn(V)e Fr(\()p Fn(x;)d(\017)p Fr(\))i Fk(\006)578 1943 y Fi(q)p 620 1943 583 2 v 53 x Fn(\014)651 1981 y Fm(2)670 1996 y Fr(\()p Fn(x;)e(\017)p Fr(\))i(+)h Fn(\015)865 1981 y Fm(2)885 1996 y Fr(\()p Fn(x;)d(\017)p Fr(\))i(+)h Fn(\016)1076 1981 y Fm(2)1095 1996 y Fr(\()p Fn(x;)d(\017)p Fr(\))647 b(\(2.1\))0 2086 y(where)16 b Fn(x)e Fk(2)g Fn(I)-10 b(R)279 2065 y Fj(n)303 2086 y Fr(,)16 b Fn(\017)e(>)g Fr(0,)i(and)h Fn(\014)s Fr(,)e Fn(\015)s Fr(,)h(and)h Fn(\016)h Fr(satisfy)e(\(1.33\).)22 b(Let)17 b Fn(a)1247 2068 y Ff(C)1269 2086 y 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Fi(\023)1531 2383 y(#)1581 2456 y Fn(b:)209 b Fr(\(2.15\))951 2753 y(15)p eop %%Page: 16 16 16 15 bop 0 -22 a Fg(Pro)r(of:)0 39 y Fr(W)l(e)17 b(\014rst)g(do)g (formal)e(p)q(erturbation)j(calculations)e(and)i(then)e(pro)o(v)o(e)g (the)h(appropriate)g(estimates.)k(W)l(e)0 99 y(b)q(egin)d(b)o(y)e (expanding)i(the)f(p)q(oten)o(tial)g Fn(V)769 81 y Ff(C)791 99 y Fr(\()p Fn(\017z)855 81 y Ff(C)877 99 y Fr(\()p Fn(s)p Fr(\))p Fn(;)8 b(\017)p Fr(\))17 b(around)i(the)e(origin)g(in)g Fn(I)-10 b(R)1515 78 y Fj(n)p Fm(+1)1601 99 y Fr(and)18 b(inserting)f(an)0 159 y Fl(a)g(priori)e Fr(expansion)i(of)f(the)g(t)o (yp)q(e)122 242 y Fn(z)147 221 y Ff(C)169 242 y Fr(\()p Fn(s)p Fr(\))e(=)g Fn(z)321 221 y Ff(C)319 254 y Fm(0)343 242 y Fr(\()p Fn(s)p Fr(\))d(+)g Fn(\017z)509 221 y Ff(C)507 254 y Fm(1)531 242 y Fr(\()p Fn(s)p Fr(\))g(+)g Fn(\017)672 221 y Fm(2)692 242 y Fn(z)717 221 y Ff(C)715 254 y Fm(2)739 242 y Fr(\()p Fn(s)p Fr(\))g(+)g Fk(\001)d(\001)g(\001)907 b Fr(\(2.16\))0 325 y(in)15 b(\(2.11\).)22 b(W)l(e)15 b(formally)e(solv)o(e)i(the)g(resulting)g(equation)h(order)f(b)o(y)g (order)h(in)f Fn(\017)p Fr(.)20 b(T)l(o)c(a)o(v)o(oid)f(con\015icts)g (in)0 394 y(notation,)i(w)o(e)e(denote)i(the)f Fn(j)545 376 y Fr(th)609 394 y(comp)q(onen)o(t)f(of)i Fn(z)939 376 y Ff(C)961 394 y Fr(\()p Fn(s)p Fr(\))f(b)o(y)g Fn(w)1142 376 y Ff(C)1141 407 y Fj(j)1165 394 y Fr(\()p Fn(s)p Fr(\).)73 454 y(F)l(rom)f(\(1.9\),)h(\(2.3\),)g(and)h(lemma)c(2.1,)j(w) o(e)g(kno)o(w)h(that)122 661 y Fn(z)147 641 y Ff(C)169 661 y Fr(\()p Fn(s)p Fr(\))d(=)296 501 y Fi(0)296 574 y(B)296 599 y(B)296 624 y(B)296 649 y(B)296 674 y(B)296 699 y(B)296 725 y(@)341 531 y Fn(\021)367 513 y Fm(0)365 543 y(1)387 531 y Fr(\()p Fn(\017)p Fr(\))381 591 y(0)341 652 y Fn(\021)367 633 y Fm(0)365 664 y(3)387 652 y Fr(\()p Fn(\017)p Fr(\))386 698 y(.)386 715 y(.)386 731 y(.)340 791 y Fn(\021)366 773 y Fm(0)364 804 y Fj(n)388 791 y Fr(\()p Fn(\017)p Fr(\))454 501 y Fi(1)454 574 y(C)454 599 y(C)454 624 y(C)454 649 y(C)454 674 y(C)454 699 y(C)454 725 y(A)515 661 y Fn(s)27 b Fr(+)g Fk(O)q Fr(\()p Fn(\017s)733 641 y Fm(2)753 661 y Fr(\))p Fn(:)1039 b Fr(\(2.17\))0 865 y(By)16 b(explicit)e(calculation,)h(w)o(e)h(see)f(that)133 955 y Fk(\000)c(r)p Fn(V)269 919 y Fe(A)271 947 y(B)300 955 y Fr(\()p Fn(\017z)369 919 y Fe(A)371 947 y(B)401 955 y Fr(\()p Fn(s)p Fr(\))p Fn(;)d(\017)p Fr(\))13 b(=)41 b Fk(\000)g(r)745 939 y Fi(e)737 955 y Fn(V)11 b Fr(\(0)p Fn(;)d Fr(0\))k(+)f Fk(O)q Fr(\(1\))615 1164 y Fk(\007)695 991 y Fi(2)695 1064 y(6)695 1089 y(6)695 1114 y(6)695 1139 y(6)695 1164 y(6)695 1189 y(6)695 1214 y(6)695 1240 y(4)723 991 y(0)723 1064 y(B)723 1089 y(B)723 1114 y(B)723 1139 y(B)723 1164 y(B)723 1189 y(B)723 1214 y(B)723 1240 y(@)788 1040 y Fn(b)809 1022 y Fm(2)809 1052 y(1)828 1040 y Fn(\017w)889 995 y Fe(A)891 1023 y(B)883 1051 y Fm(1)921 1040 y Fr(\()p Fn(s)p Fr(\))768 1119 y Fn(b)789 1126 y Fm(1)808 1119 y Fn(b)829 1126 y Fm(2)849 1119 y Fn(\017w)910 1074 y Fe(A)912 1102 y(B)904 1130 y 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y Fm(2)p 127 1887 69 2 v 127 1933 a Fn(ds)175 1918 y Fm(2)208 1899 y Fn(z)233 1878 y Ff(C)231 1911 y Fm(1)256 1899 y Fr(\()p Fn(s)p Fr(\))30 b(=)g Fk(\000r)504 1882 y Fi(e)496 1899 y Fn(V)10 b Fr(\(0)p Fn(;)e Fr(0\))20 b Fk(\007)884 1865 y Fn(\021)910 1847 y Fm(0)908 1877 y(1)930 1865 y Fr(\()p Fn(\017)p Fr(\))8 b Fn(s)p 725 1887 453 2 v 725 1895 a Fi(q)p 767 1895 412 2 v 51 x Fr(\()p Fn(\021)812 1929 y Fm(0)810 1957 y(1)831 1946 y Fr(\()p Fn(\017)p Fr(\))p Fn(s)p Fr(\))931 1931 y Fm(2)962 1946 y Fr(+)j(\()p Fn(d)1055 1953 y Fm(3)1075 1946 y Fn(=b)1120 1953 y Fm(1)1140 1946 y Fr(\))1159 1931 y Fm(2)1192 1899 y Fn(b:)598 b Fr(\(2.19\))0 2038 y(In)o(tegrating)16 b(and)h(using)479 2018 y Fj(d)p 479 2026 35 2 v 479 2055 a(ds)519 2038 y Fn(z)544 2020 y Ff(C)542 2050 y Fm(1)566 2038 y Fr(\(0\))d(=)g(0,)i(w)o(e)g(obtain)127 2123 y Fn(d)p 127 2145 49 2 v 127 2191 a(ds)189 2157 y(z)214 2136 y Ff(C)212 2169 y Fm(1)236 2157 y Fr(\()p Fn(s)p Fr(\))30 b(=)g Fk(\000r)484 2141 y Fi(e)476 2157 y 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Fr(W)l(e)c(assume)f(from)g(no)o(w)i(on)f(that)h(the)f(solution)g Fn(a)p Fr(\()p Fn(t)p Fr(\))f(of)i(\(2.2\))f(with)g Fn(V)1329 53 y Ff(C)1352 71 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))13 b Fk(\021)1534 55 y Fi(e)1526 71 y Fn(V)e Fr(\()p Fn(x;)d(\017)p Fr(\))15 b(is)h(sub)s(ject)g(to)0 131 y(the)g(initial)f(conditions)122 233 y Fn(a)p Fr(\(0\))42 b(=)f(0)1470 b(\(2.34\))122 305 y Fn(\021)r Fr(\(0\))42 b(=)f Fn(\021)357 285 y Fm(0)0 407 y Fr(whereas)17 b(the)f(solutions)g Fn(a)498 389 y Ff(C)520 407 y Fr(\()p Fn(t)p Fr(\))g(satisfy)122 509 y Fn(a)148 488 y Ff(C)170 509 y Fr(\(0\))42 b(=)g(0)1447 b(\(2.35\))122 581 y Fn(\021)148 561 y Ff(C)170 581 y Fr(\(0\))42 b(=)g Fn(\021)380 561 y Fm(0)399 581 y Fr(\()p Fn(\017)p Fr(\))14 b(=)f Fn(\021)548 561 y Fm(0)579 581 y Fr(+)e Fk(O)q Fr(\()p Fn(\017)p Fr(\))p Fn(:)0 683 y Fr(The)21 b Fk(O)q Fr(\()p Fn(\017)p Fr(\))f(term)f(m)o(ust)g(b)q(e)h (included)g(in)g(our)h(calculations)f(b)q(ecause)h(when)g(the)f (electrons)g(mak)o(es)f(a)0 743 y(transition)e(from)f(one)h(energy)f (lev)o(el)f(surface)i(to)g(another,)g(the)f(n)o(uclei)f(m)o(ust)h(comp) q(ensate)g(b)o(y)g(making)0 803 y(a)h(c)o(hange)f(in)g(their)f(kinetic) g(energy)h(in)g(order)g(to)g(conserv)o(e)g(the)g(total)g(energy)g(of)h (the)f(whole)g(system.)73 864 y(W)l(e)g(easily)g(get)g(the)g(estimates) 0 978 y Fg(Corollary)i(2.2)24 b Fl(When)18 b Fn(\017)13 b Fk(!)h Fr(0)p Fl(,)k Fn(t)13 b Fk(!)h Fr(0)p Fl(,)k Fn(t)810 960 y Fm(3)829 978 y Fn(=\017)873 960 y Fm(2)907 978 y Fk(!)13 b Fr(0)18 b Fl(and)g Fk(j)p Fn(t)p Fk(j)p Fn(=\017)13 b Fk(!)g(1)18 b Fl(we)g(have)243 1089 y Fn(\021)274 1053 y Fe(A)276 1081 y(B)305 1089 y Fr(\()p Fn(t)p Fr(\)\()p Fn(a)p Fr(\()p Fn(t)p Fr(\))10 b Fk(\000)h Fn(a)553 1053 y Fe(A)555 1081 y(B)584 1089 y Fr(\()p Fn(t)p Fr(\)\))163 1161 y(=)42 b Fk(\007j)p Fn(d)321 1168 y Fm(3)341 1161 y Fk(j)p Fn(\017t)10 b Fk(\000)h Fr(\()p Fn(\021)498 1141 y Fm(0)517 1161 y Fr(\()p Fn(\017)p Fr(\))g Fk(\000)g Fn(\021)662 1141 y Fm(0)681 1161 y Fr(\))p Fn(\021)726 1141 y Fm(0)746 1161 y Fr(\()p Fn(\017)p Fr(\))p Fn(t)243 1265 y Fk(\006)p Fr(sign\()p Fn(t)p Fr(\))430 1192 y Fi(")459 1231 y Fn(b)480 1238 y Fm(1)499 1231 y Fn(\021)525 1213 y Fm(0)523 1244 y(1)545 1231 y Fr(\()p Fn(\017)p Fr(\))p 459 1253 144 2 v 519 1299 a(2)608 1265 y Fn(t)626 1245 y Fm(2)656 1265 y Fr(+)752 1231 y Fn(\017)772 1213 y Fm(2)791 1231 y Fn(d)816 1213 y Fm(2)816 1244 y(3)p 710 1253 169 2 v 710 1299 a Fr(2)p Fn(b)755 1306 y Fm(1)775 1299 y Fn(\021)801 1282 y Fm(0)799 1310 y(1)820 1299 y Fr(\()p Fn(\017)p Fr(\))891 1192 y Fi( )924 1265 y Fr(ln)973 1192 y Fi( )1011 1231 y Fr(2)p Fn(b)1056 1238 y Fm(1)1076 1231 y Fn(\021)1102 1213 y Fm(0)1100 1244 y(1)1121 1231 y Fr(\()p Fn(\017)p Fr(\))p Fk(j)p Fn(t)p Fk(j)p 1011 1253 214 2 v 1071 1299 a(j)p Fn(d)1110 1306 y Fm(3)1130 1299 y Fk(j)p Fn(\017)1229 1192 y Fi(!)1273 1265 y Fr(+)1327 1231 y(1)p 1327 1253 25 2 v 1327 1299 a(2)1357 1192 y Fi(!)o(#)243 1373 y Fr(+)p Fk(O)q Fr(\()p Fn(t)359 1353 y Fm(3)378 1373 y Fr(\))g(+)g Fk(O)q Fr(\()p Fn(\017)537 1353 y Fm(4)557 1373 y Fn(=t)599 1353 y Fm(2)618 1373 y Fr(\))g(+)g Fk(O)q Fr(\()p Fn(t\017)795 1353 y Fm(2)823 1373 y Fr(ln)d Fn(\017)p Fr(\))0 1488 y(and)0 1589 y Fg(Corollary)18 b(2.3)24 b Fl(When)18 b Fn(\017)13 b Fk(!)h Fr(0)p Fl(,)k Fn(t)13 b Fk(!)h Fr(0)p Fl(,)k Fn(t)810 1571 y Fm(3)829 1589 y Fn(=\017)873 1571 y Fm(2)907 1589 y Fk(!)13 b Fr(0)18 b Fl(and)g Fk(j)p Fn(t)p Fk(j)p Fn(=\017)13 b Fk(!)g(1)18 b Fl(we)g(have)122 1712 y Fn(S)160 1676 y Fe(A)162 1704 y(B)192 1712 y Fr(\()p Fn(t)p Fr(\))41 b(=)g Fn(S)406 1666 y Fe(A)408 1695 y(B)398 1723 y Fm(0)438 1712 y Fr(\()p Fn(\017;)8 b Fr(sign)o(\()p Fn(t)p Fr(\)\))j(+)g Fn(S)s Fr(\()p Fn(t)p Fr(\))g(+)g(\()p Fn(\021)911 1691 y Fm(0)930 1683 y(2)950 1712 y Fr(\()p Fn(\017)p Fr(\))f Fk(\000)h Fn(\021)1094 1691 y Fm(0)1114 1683 y(2)1134 1712 y Fr(\))1161 1678 y Fn(t)p 1158 1700 V 1158 1746 a Fr(2)1198 1712 y Fk(\006)g Fn(\017)p Fk(j)p Fn(d)1307 1719 y Fm(3)1326 1712 y Fk(j)p Fn(t)368 1837 y Fk(\007)p Fr(sign\()p Fn(t)p Fr(\))555 1764 y Fi( )588 1837 y Fn(\021)614 1817 y Fm(0)612 1850 y(1)633 1837 y Fr(\()p Fn(\017)p Fr(\))p Fn(b)712 1844 y Fm(1)731 1837 y Fn(t)749 1817 y Fm(2)780 1837 y Fr(+)863 1804 y Fn(d)888 1786 y Fm(2)888 1816 y(3)908 1804 y Fn(\017)928 1786 y Fm(2)p 834 1826 144 2 v 834 1871 a Fn(\021)860 1854 y Fm(0)858 1882 y(1)879 1871 y Fr(\()p Fn(\017)p Fr(\))p Fn(b)958 1878 y Fm(1)999 1837 y Fr(ln)16 b Fk(j)p Fn(t)p Fk(j)1102 1764 y Fi(!)368 1946 y Fr(+)p Fk(O)q Fr(\()p Fn(t)484 1925 y Fm(3)503 1946 y Fr(\))11 b(+)g Fk(O)q Fr(\()p Fn(\017)662 1925 y Fm(4)682 1946 y Fn(=t)724 1925 y Fm(2)743 1946 y Fr(\))h(+)f Fk(O)q Fr(\()p Fn(\017)903 1925 y Fm(3)930 1946 y Fr(ln)d Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(:)0 2132 y Fc(2.6)70 b(Matrices)21 b Fb(A)486 2110 y Fu(C)513 2132 y Fv(\()p Fb(t)p Fv(\))i Fc(and)g Fb(B)784 2110 y Fu(C)811 2132 y Fv(\()p Fb(t)p Fv(\))0 2224 y Fr(The)13 b(semiclassical)e(w)o(a)o(v)o 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Fm(0)365 71 y Fn(;)1446 b Fr(\(2.37\))127 143 y Fn(B)167 123 y Ff(C)189 143 y Fr(\(0\))14 b(=)g Fn(B)354 150 y Fm(0)374 143 y Fn(;)0 236 y Fr(where)k Fn(a)169 218 y Ff(C)191 236 y Fr(\()p Fn(t)p Fr(\))f(is)h(the)g(solution)g(of)g (\(2.2\))h(and)f(\(2.3\).)27 b(Let)18 b(us)g(determine)e(the)h(leading) h(order)g(b)q(eha)o(vior)0 296 y(of)f Fn(V)95 278 y Fm(\(2\))142 296 y Fr(\()p Fn(a)187 278 y Ff(C)209 296 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(\017)p Fr(\).)20 b(By)c(explicit)e(computation)i(w)o (e)f(get,)h(\(omitting)f(the)h(argumen)o(ts)f(\()p Fn(x;)8 b(\017)p Fr(\)\))122 388 y Fn(V)161 368 y Fm(\(2\))222 388 y Fr(=)351 372 y Fi(e)343 388 y Fn(V)382 368 y Fm(\(2\))441 388 y Fk(\006)i Fr(\()p Fk(jr)p Fn(\014)s Fk(ihr)p Fn(\014)s Fk(j)g Fr(+)h Fk(jr)p Fn(\015)s Fk(ihr)p Fn(\015)s Fk(j)f Fr(+)h Fk(jr)p Fn(\016)r Fk(ihr)p Fn(\016)r Fk(j)p Fr(\))c(\()p Fn(\014)1319 368 y Fm(2)1349 388 y Fr(+)k Fn(\015)1426 368 y Fm(2)1457 388 y Fr(+)g Fn(\016)1530 368 y Fm(2)1549 388 y Fr(\))1568 368 y Ff(\000)p Fm(1)p Fj(=)p Fm(2)343 461 y Fr(+\()p Fn(\014)s(\014)462 440 y Fm(\(2\))519 461 y Fr(+)g Fn(\015)s(\015)624 440 y Fm(\(2\))682 461 y Fr(+)g Fn(\016)r(\016)779 440 y Fm(\(2\))825 461 y Fr(\)\()p Fn(\014)894 440 y Fm(2)924 461 y Fr(+)g Fn(\015)1001 440 y Fm(2)1032 461 y Fr(+)g Fn(\016)1105 440 y Fm(2)1124 461 y Fr(\))1143 440 y Ff(\000)p Fm(1)p Fj(=)p Fm(2)343 533 y Fk(\000j)p Fn(\014)s Fk(r)p Fn(\014)h Fr(+)f Fn(\015)s Fk(r)p Fn(\015)j Fr(+)d Fn(\016)r Fk(r)p Fn(\016)r Fk(ih)p Fn(\014)s Fk(r)p Fn(\014)g Fr(+)g Fn(\015)s Fk(r)p Fn(\015)j Fr(+)d Fn(\016)r Fk(r)p Fn(\016)r Fk(j)p Fr(\()p Fn(\014)1317 513 y Fm(2)1345 533 y Fr(+)g Fn(\015)1422 513 y Fm(2)1453 533 y Fr(+)g Fn(\016)1526 513 y Fm(2)1545 533 y Fr(\))1564 513 y Ff(\000)p Fm(3)p Fj(=)p Fm(2)1646 533 y Fn(:)165 b Fr(\(2.38\))0 626 y(Th)o(us,)16 b(from)f(\(1.33\))i(and)g(some)e (simple)f(estimates,)g(w)o(e)i(see)g(that)205 718 y Fn(V)244 697 y Fm(\(2\))321 718 y Fr(=)398 702 y Fi(e)390 718 y Fn(V)429 697 y Fm(\(2\))1825 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y(7)1915 1574 y(7)1915 1600 y(5)1951 1536 y Fn(:)0 1747 y Fr(Using)16 b(this)g(expression,)g (w)o(e)f(obtain)i(the)f(follo)o(wing)g(result)g(for)g(the)g(solutions)h (to)g(\(2.36\))g(and)g(\(2.37\):)0 1849 y Fg(Prop)r(osition)h(2.3)24 b Fl(L)n(et)18 b Fn(A)521 1830 y Ff(C)543 1849 y Fr(\()p Fn(t)p Fr(\))g Fl(and)g Fn(B)752 1830 y Ff(C)774 1849 y Fr(\()p Fn(t)p Fr(\))g Fl(b)n(e)h(the)f(solutions)h(of)g(\(2.36\))e (and)i(\(2.37\).)24 b(In)19 b(the)f(r)n(e)n(gime)0 1909 y Fn(\017)c Fk(!)f Fr(0)p Fl(,)18 b Fk(j)p Fn(t)p Fk(j)13 b(!)h Fr(0)p Fl(,)j Fn(\017=)p Fk(j)p Fn(t)p Fk(j)d(!)f Fr(0)p Fl(,)18 b(and)g Fk(j)p Fn(t)e Fr(ln)740 1887 y Fm(2)777 1909 y Fn(\017)8 b Fk(j)13 b(!)h Fr(0)p Fl(,)k(we)g(have)g (the)g(fol)r(lowing)i(asymptotics:)125 2010 y Fn(A)167 1974 y Fe(A)168 2003 y(B)198 2010 y Fr(\()p Fn(t)p Fr(\))41 b(=)h Fn(A)412 2017 y Fm(0)459 2010 y Fr(+)29 b Fk(O)9 b Fr(\()p Fk(j)p Fn(t)p Fk(j)f(j)g Fr(ln)16 b Fn(\017)p Fk(j)p Fr(\))8 b Fn(;)1023 b Fr(\(2.41\))122 2114 y Fn(B)166 2078 y Fe(A)168 2106 y(B)198 2114 y Fr(\()p Fn(t)p Fr(\))41 b(=)h Fn(B)412 2121 y Fm(0)451 2114 y Fk(\006)514 2080 y Fn(i)8 b Fr(sign\()p Fn(t)p Fr(\))p 514 2102 165 2 v 555 2148 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))692 2041 y Fi(")716 2114 y Fn(M)d(A)805 2121 y Fm(0)844 2114 y Fr(+)20 b(ln)959 2041 y Fi( )997 2080 y Fr(2)p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))p Fk(j)p Fn(t)p Fk(j)p 997 2102 153 2 v 1063 2148 a Fn(\017)1154 2041 y Fi(!)1195 2114 y Fn(N)5 b(A)1276 2121 y Fm(0)1296 2041 y Fi(#)1339 2114 y Fr(+)20 b Fk(O)1446 2041 y Fi( )1479 2114 y Fk(j)p Fn(t)p Fk(j)c Fr(ln)1581 2093 y Fm(2)1618 2114 y Fn(\017)j Fr(+)1711 2080 y Fn(\017)1731 2062 y Fm(2)p 1711 2102 40 2 v 1712 2148 a Fn(t)1730 2134 y Fm(2)1755 2041 y Fi(!)1796 2114 y Fn(;)1825 2218 y Fr(\(2.42\))0 2310 y Fl(wher)n(e)122 2508 y Fn(M)27 b Fr(=)286 2475 y(1)p 262 2497 73 2 v 262 2542 a Fk(j)p Fn(d)301 2549 y Fm(3)320 2542 y Fk(j)339 2508 y(j)p Fn(b)p Fk(ih)p Fn(b)p Fk(j)22 b Fr(=)529 2348 y Fi(0)529 2421 y(B)529 2446 y(B)529 2471 y(B)529 2496 y(B)529 2521 y(B)529 2546 y(B)529 2572 y(@)594 2384 y Fn(b)615 2366 y Fm(2)615 2397 y(1)635 2384 y Fn(=)p Fk(j)p Fn(d)698 2391 y Fm(3)718 2384 y Fk(j)70 b Fn(b)823 2391 y Fm(1)842 2384 y Fn(b)863 2391 y Fm(2)883 2384 y Fn(=)p Fk(j)p Fn(d)946 2391 y Fm(3)966 2384 y Fk(j)50 b Fr(0)i Fn(:)8 b(:)g(:)51 b Fr(0)574 2445 y Fn(b)595 2452 y Fm(1)614 2445 y Fn(b)635 2452 y Fm(2)655 2445 y Fn(=)p Fk(j)p Fn(d)718 2452 y Fm(3)738 2445 y Fk(j)70 b Fn(b)843 2426 y Fm(2)843 2457 y(2)863 2445 y Fn(=)p Fk(j)p Fn(d)926 2452 y Fm(3)946 2445 y Fk(j)g Fr(0)52 b Fn(:)8 b(:)g(:)51 b Fr(0)651 2505 y(0)204 b(0)127 b(0)52 b Fn(:)8 b(:)g(:)51 b Fr(0)656 2539 y Fl(.)656 2555 y(.)656 2572 y(.)883 2539 y(.)883 2555 y(.)883 2572 y(.)1034 2539 y(.)1034 2555 y(.)1034 2572 y(.)1107 2543 y(.)1127 2555 y(.)1148 2568 y(.)1220 2539 y(.)1220 2555 y(.)1220 2572 y(.)651 2632 y Fr(0)204 b(0)127 b(0)52 b Fn(:)8 b(:)g(:)51 b Fr(0)1248 2348 y Fi(1)1248 2421 y(C)1248 2446 y(C)1248 2471 y(C)1248 2496 y(C)1248 2521 y(C)1248 2546 y(C)1248 2572 y(A)1825 2508 y Fr(\(2.43\))951 2753 y(20)p eop %%Page: 21 21 21 20 bop 0 -22 a Fl(and)122 186 y Fn(N)27 b Fr(=)278 152 y(1)p 253 174 73 2 v 253 220 a Fk(j)p Fn(d)292 227 y Fm(3)312 220 y Fk(j)331 186 y(j)p Fn(c)p Fk(ih)p Fn(c)p Fk(j)c Fr(=)522 26 y Fi(0)522 99 y(B)522 124 y(B)522 149 y(B)522 174 y(B)522 198 y(B)522 223 y(B)522 250 y(@)566 62 y Fr(0)107 b(0)g(0)52 b Fn(:)8 b(:)g(:)52 b Fr(0)566 122 y(0)f Fn(c)662 104 y Fm(2)662 135 y(2)681 122 y Fn(=)p Fk(j)p Fn(d)744 129 y Fm(3)765 122 y Fk(j)e Fr(0)j Fn(:)8 b(:)g(:)52 b Fr(0)566 183 y(0)107 b(0)g(0)52 b Fn(:)8 b(:)g(:)52 b Fr(0)571 216 y Fl(.)571 233 y(.)571 250 y(.)702 216 y(.)702 233 y(.)702 250 y(.)833 216 y(.)833 233 y(.)833 250 y(.)905 221 y(.)926 233 y(.)946 246 y(.)1018 216 y(.)1018 233 y(.)1018 250 y(.)566 310 y Fr(0)107 b(0)g(0)52 b Fn(:)8 b(:)g(:)52 b Fr(0)1046 26 y Fi(1)1046 99 y(C)1046 124 y(C)1046 149 y(C)1046 174 y(C)1046 198 y(C)1046 223 y(C)1046 250 y(A)1091 186 y Fn(:)720 b Fr(\(2.44\))0 416 y Fg(Pro)r(of:)0 476 y Fr(T)l(o)12 b(study)g(solutions)g(to)g (\(2.36\))g(and)h(\(2.37\))f(w)o(e)f(\014rst)h(consider)f(v)m(arious)i (quan)o(tities)d(that)i(o)q(ccur)g(in)f(\(2.40\))0 536 y(with)17 b Fn(x)f Fr(replaced)g(b)o(y)g Fn(a)443 518 y Ff(C)465 536 y Fr(\()p Fn(t)p Fr(\))e(=)h Fn(\021)614 518 y Fm(0)633 536 y Fr(\()p Fn(\017)p Fr(\))p Fn(t)c Fr(+)g Fk(O)q Fr(\()p Fn(t)847 518 y Fm(2)866 536 y Fr(\).)23 b(By)16 b(tedious,)g(but)h(straigh)o(tforw)o(ard)h(calculations,)d(w)o (e)0 596 y(obtain)i(the)f(follo)o(wing)g(estimates)227 691 y Fn(\014)258 673 y Fm(2)288 691 y Fr(+)11 b Fn(\016)361 673 y Fm(2)p 127 713 355 2 v 127 759 a Fr(\()p Fn(\014)177 745 y Fm(2)207 759 y Fr(+)g Fn(\015)284 745 y Fm(2)315 759 y Fr(+)g Fn(\016)388 745 y Fm(2)407 759 y Fr(\))426 745 y Fm(3)p Fj(=)p Fm(2)527 725 y Fr(=)781 691 y(1)p 612 713 362 2 v 612 771 a Fk(j)p Fn(d)651 778 y Fm(3)671 771 y Fk(j)693 721 y Fi(q)p 734 721 240 2 v 734 771 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))817 757 y Fm(2)837 771 y Fn(t)855 757 y Fm(2)885 771 y Fr(+)g Fn(\017)954 757 y Fm(2)990 725 y Fr(+)19 b Fk(O)1096 677 y Fi(\020)1121 725 y Fr(\()p Fn(t)1158 704 y Fm(2)1189 725 y Fr(+)11 b Fn(\017)1258 704 y Fm(2)1277 725 y Fr(\))1296 704 y Fm(0)1316 677 y Fi(\021)228 851 y Fn(\015)256 832 y Fm(2)287 851 y Fr(+)g Fn(\016)360 832 y Fm(2)p 127 873 355 2 v 127 918 a Fr(\()p Fn(\014)177 904 y Fm(2)207 918 y Fr(+)g Fn(\015)284 904 y Fm(2)315 918 y Fr(+)g Fn(\016)388 904 y Fm(2)407 918 y Fr(\))426 904 y Fm(3)p Fj(=)p Fm(2)527 884 y Fr(=)799 851 y Fn(\017)819 832 y Fm(2)p 612 873 414 2 v 612 918 a Fk(j)p Fn(d)651 925 y Fm(3)671 918 y Fk(j)d Fr(\()p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))795 904 y Fm(2)814 918 y Fn(t)832 904 y Fm(2)863 918 y Fr(+)j Fn(\017)932 904 y Fm(2)951 918 y Fr(\))970 904 y Fm(3)p Fj(=)p Fm(2)1041 884 y Fr(+)20 b Fk(O)1148 836 y Fi(\020)1173 884 y Fr(\()p Fn(t)1210 864 y Fm(2)1240 884 y Fr(+)11 b Fn(\017)1309 864 y Fm(2)1328 884 y Fr(\))1347 864 y Fm(0)1367 836 y Fi(\021)271 982 y Fn(\014)f(\015)p 127 1004 355 2 v 127 1050 a Fr(\()p Fn(\014)177 1036 y Fm(2)207 1050 y Fr(+)h Fn(\015)284 1036 y Fm(2)315 1050 y Fr(+)g Fn(\016)388 1036 y Fm(2)407 1050 y Fr(\))426 1036 y Fm(3)p Fj(=)p Fm(2)527 1016 y Fr(=)42 b Fk(O)656 968 y Fi(\020)681 1016 y Fr(\()p Fn(t)718 995 y Fm(2)748 1016 y Fr(+)11 b Fn(\017)817 995 y Fm(2)837 1016 y Fr(\))856 995 y Fm(0)875 968 y Fi(\021)909 1016 y Fn(:)0 1143 y Fr(F)l(rom)k(these)h(estimates,) e(it)i(follo)o(ws)g(that)h(for)f(small)f Fn(t)h Fr(and)h Fn(\017)p Fr(,)122 1273 y Fn(V)161 1252 y Fm(\(2\))208 1273 y Fr(\()p Fn(a)258 1237 y Fe(A)260 1265 y(B)290 1273 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(\017)p Fr(\))23 b(=)i Fk(\006)540 1200 y Fi( )724 1239 y Fn(\017)744 1221 y Fm(2)p 578 1261 333 2 v 578 1307 a Fr(\()p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))680 1293 y Fm(2)699 1307 y Fn(t)717 1293 y Fm(2)748 1307 y Fr(+)11 b Fn(\017)817 1293 y Fm(2)836 1307 y Fr(\))855 1293 y Fm(3)p Fj(=)p Fm(2)923 1273 y Fn(M)25 b Fr(+)1211 1239 y(1)p 1057 1261 V 1057 1307 a(\()p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1159 1293 y Fm(2)1179 1307 y Fn(t)1197 1293 y Fm(2)1227 1307 y Fr(+)11 b Fn(\017)1296 1293 y Fm(2)1316 1307 y Fr(\))1335 1293 y Fm(1)p Fj(=)p Fm(2)1403 1273 y Fn(N)1447 1200 y Fi(!)1488 1273 y Fr(+)d Fk(O)1584 1225 y Fi(\020)1609 1273 y Fr(\()p Fn(t)1646 1252 y Fm(2)1676 1273 y Fr(+)j Fn(\017)1745 1252 y Fm(2)1765 1273 y Fr(\))1784 1252 y Fm(0)1803 1225 y Fi(\021)1836 1273 y Fn(:)p Fr(\(2.45\))73 1406 y(W)l(e)16 b(can)h(solv)o(e)e(\(2.36\))i(and)g(\(2.37\))g(if)e(and)i(only)f(if)g (w)o(e)g(can)g(solv)o(e)122 1530 y Fn(A)159 1509 y Ff(C)181 1530 y Fr(\()p Fn(t;)8 b(\017)p Fr(\))13 b(=)h Fn(A)381 1537 y Fm(0)411 1530 y Fr(+)d Fn(itB)532 1537 y Fm(0)562 1530 y Fk(\000)612 1471 y Fi(Z)654 1484 y Fj(t)635 1565 y Fm(0)685 1471 y Fi(Z)726 1484 y Fj(s)708 1565 y Fm(0)761 1530 y Fn(V)801 1509 y Fm(\(2\))848 1530 y Fr(\()p Fn(a)893 1509 y Ff(C)915 1530 y Fr(\()p Fn(r)o(;)d(\017)p Fr(\)\))g Fn(A)1080 1509 y Ff(C)1101 1530 y Fr(\()p Fn(r)o(;)g(\017)p Fr(\))g Fn(dr)o(:)555 b Fr(\(2.46\))0 1648 y(F)l(or)16 b(eac)o(h)g(\014xed)g(p)q(ositiv)o(e)f Fn(\017)p Fr(,)h(this)g (equation)g(can)h(b)q(e)f(solv)o(ed)g(b)o(y)f(iteration)h(for)h(small)d Fk(j)p Fn(t)p Fk(j)p Fr(.)73 1709 y(F)l(or)j Fn(T)j(>)14 b Fr(0,)i(w)o(e)g(de\014ne)f(a)i(norm)e(on)i(b)q(ounded)g(op)q(erator)h (v)m(alued)e(functions)g(of)h Fn(t)f Fr(and)g Fn(\017)g Fr(b)o(y)122 1810 y Fk(jjj)8 b Fn(D)q Fr(\()p Fk(\001)p Fn(;)g Fk(\001)p Fr(\))g Fk(jjj)30 b Fr(=)210 b(sup)449 1852 y Ff(f)p Fj(\017)p Ff(\024)p Fj(T)s(;)6 b Ff(j)p Fj(t)11 b Fm(ln)g Fj(t)p Ff(j\024)p Fj(T)s(;)5 b Ff(j)p Fj(t)11 b Fm(ln)h Fj(\017)p Ff(j\024)p Fj(T)5 b Ff(g)899 1810 y Fk(k)j Fn(D)q Fr(\()p Fn(t;)g(\017)p Fr(\))g Fk(k)p Fn(:)0 1947 y Fr(F)l(rom)15 b(\(2.46\),)h(w)o(e)g(see)g(that)h(for)f(p) q(ositiv)o(e)g Fn(\017)p Fr(,)122 2071 y Fk(k)p Fn(A)184 2051 y Ff(C)206 2071 y Fr(\()p Fn(t;)8 b(\017)p Fr(\))i Fk(\000)h Fn(A)401 2078 y Fm(0)420 2071 y Fk(k)42 b Fr(=)567 2009 y Fi(\015)567 2034 y(\015)567 2059 y(\015)567 2084 y(\015)598 2071 y Fn(itB)670 2078 y Fm(0)700 2071 y Fk(\000)750 2013 y Fi(Z)792 2026 y Fj(t)773 2107 y Fm(0)815 2071 y Fn(ds)871 2013 y Fi(Z)913 2026 y Fj(s)894 2107 y Fm(0)940 2071 y Fn(V)979 2051 y Fm(\(2\))1026 2071 y Fr(\()p Fn(a)1071 2051 y Ff(C)1093 2071 y Fr(\()p Fn(r)o(;)8 b(\017)p Fr(\)\))p Fn(A)1250 2051 y Ff(C)1271 2071 y Fr(\()p Fn(r)o(;)g(\017)p Fr(\))g Fn(dr)1437 2009 y Fi(\015)1437 2034 y(\015)1437 2059 y(\015)1437 2084 y(\015)487 2191 y Fk(\024)41 b(j)p Fn(t)p Fk(j)8 b(k)p Fn(B)683 2198 y Fm(0)702 2191 y Fk(k)j Fr(+)787 2129 y Fi(\014)787 2154 y(\014)787 2179 y(\014)787 2203 y(\014)809 2132 y(Z)851 2146 y Fj(t)832 2227 y Fm(0)874 2191 y Fn(ds)930 2132 y Fi(Z)972 2146 y Fj(s)954 2227 y Fm(0)1007 2191 y Fk(k)p Fn(V)1071 2170 y Fm(\(2\))1118 2191 y Fr(\()p Fn(a)1163 2170 y Ff(C)1185 2191 y Fr(\()p Fn(r)o(;)d(\017)p Fr(\)\))p Fk(k)g(jjj)p Fn(A)1417 2170 y Ff(C)1438 2191 y Fr(\()p Fk(\001)p Fn(;)g Fk(\001)p Fr(\))j Fk(\000)g Fn(A)1624 2198 y Fm(0)1643 2191 y Fk(jjj)d Fn(dr)1750 2129 y Fi(\014)1750 2154 y(\014)1750 2179 y(\014)1750 2203 y(\014)1825 2191 y Fr(\(2.47\))860 2311 y(+)914 2249 y Fi(\014)914 2273 y(\014)914 2298 y(\014)914 2323 y(\014)936 2252 y(Z)978 2265 y Fj(t)959 2347 y Fm(0)1009 2311 y Fn(ds)1074 2252 y Fi(Z)1116 2265 y Fj(s)1097 2347 y Fm(0)1151 2311 y Fk(k)p Fn(V)1215 2290 y Fm(\(2\))1262 2311 y Fr(\()p Fn(a)1307 2290 y Ff(C)1329 2311 y Fr(\()p Fn(r)o(;)g(\017)p Fr(\)\))p Fk(k)15 b(k)p Fn(A)1551 2318 y Fm(0)1570 2311 y Fk(k)i Fn(dr)1669 2249 y Fi(\014)1669 2273 y(\014)1669 2298 y(\014)1669 2323 y(\014)487 2402 y Fk(\024)41 b(j)p Fn(t)p Fk(j)8 b(k)p Fn(B)683 2409 y Fm(0)702 2402 y Fk(k)583 2497 y Fr(+)646 2435 y Fi(\014)646 2460 y(\014)646 2485 y(\014)646 2510 y(\014)668 2438 y(Z)709 2452 y Fj(t)691 2533 y Fm(0)732 2497 y Fn(ds)788 2438 y Fi(Z)831 2452 y Fj(s)812 2533 y Fm(0)866 2497 y Fk(k)p Fn(V)930 2476 y Fm(\(2\))977 2497 y Fr(\()p Fn(a)1022 2476 y Ff(C)1044 2497 y Fr(\()p Fn(r)o(;)g(\017)p Fr(\)\))p Fk(k)g Fn(dr)1253 2435 y Fi(\014)1253 2460 y(\014)1253 2485 y(\014)1253 2510 y(\014)1284 2449 y(\020)1317 2497 y Fk(jjj)p Fn(A)1396 2476 y Ff(C)1417 2497 y Fr(\()p Fk(\001)p Fn(;)g Fk(\001)p Fr(\))j Fk(\000)g Fn(A)1603 2504 y Fm(0)1622 2497 y Fk(jjj)g Fr(+)g Fk(k)p Fn(A)1786 2504 y Fm(0)1805 2497 y Fk(k)1838 2449 y Fi(\021)1871 2497 y Fn(:)951 2753 y Fr(21)p eop %%Page: 22 22 22 21 bop 73 -22 a Fr(F)l(rom)15 b(\(2.45\))i(and)g(explicit)d (calculation,)h(w)o(e)h(see)g(that)g(for)h(p)q(ositiv)o(e)e Fn(\017)p Fr(,)244 12 y Fi(\014)244 37 y(\014)244 62 y(\014)244 87 y(\014)274 16 y(Z)315 29 y Fj(s)297 110 y Fm(0)358 74 y Fk(k)p Fn(V)422 54 y Fm(\(2\))470 74 y Fr(\()p Fn(a)515 54 y Ff(C)537 74 y Fr(\()p Fn(r)o(;)8 b(\017)p Fr(\)\))p Fk(k)15 b Fn(dr)762 12 y Fi(\014)762 37 y(\014)762 62 y(\014)762 87 y(\014)163 202 y Fk(\024)42 b Fn(C)279 209 y Fm(1)298 202 y Fk(j)p Fn(s)p Fk(j)19 b Fr(+)h Fn(C)461 209 y Fm(2)497 143 y Fi(Z)538 156 y Ff(j)p Fj(s)p Ff(j)520 237 y Fm(0)723 168 y Fn(\017)743 150 y Fm(2)778 168 y Fn(dr)p 606 190 338 2 v 606 236 a Fr(\()p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))708 221 y Fm(2)727 236 y Fn(r)750 221 y Fm(2)782 236 y Fr(+)11 b Fn(\017)851 221 y Fm(2)870 236 y Fr(\))889 221 y Fm(3)p Fj(=)p Fm(2)968 202 y Fr(+)20 b Fn(C)1061 209 y Fm(3)1097 143 y Fi(Z)1138 156 y Ff(j)p Fj(s)p Ff(j)1120 237 y Fm(0)1325 168 y Fn(dr)p 1206 190 287 2 v 1206 198 a Fi(q)p 1247 198 246 2 v 1247 248 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1330 234 y Fm(2)1350 248 y Fn(r)1373 234 y Fm(2)1404 248 y Fr(+)11 b Fn(\017)1473 234 y Fm(2)1825 202 y Fr(\(2.48\))164 375 y(=)42 b Fn(C)279 382 y Fm(1)298 375 y Fk(j)p Fn(s)p Fk(j)19 b Fr(+)h Fn(C)461 382 y Fm(2)611 341 y Fk(j)p Fn(s)p Fk(j)p 494 363 287 2 v 494 372 a Fi(q)p 535 372 245 2 v 535 422 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))618 407 y Fm(2)638 422 y Fn(s)661 407 y Fm(2)691 422 y Fr(+)11 b Fn(\017)760 407 y Fm(2)804 375 y Fr(+)19 b Fn(C)896 382 y Fm(3)958 341 y Fr(1)p 929 363 83 2 v 929 409 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1034 375 y(ln)1091 290 y Fi(0)1091 364 y(@)1132 341 y Fn(\032)p Fr(\()p Fn(\017)p Fr(\))p Fk(j)p Fn(s)p Fk(j)p 1132 363 134 2 v 1189 409 a Fn(\017)1282 375 y Fr(+)1331 292 y Fi(s)p 1372 292 240 2 v 1377 341 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1460 327 y Fm(2)1480 341 y Fn(s)1503 327 y Fm(2)p 1377 363 146 2 v 1430 409 a Fn(\017)1450 395 y Fm(2)1538 375 y Fr(+)11 b(1)1612 290 y Fi(1)1612 364 y(A)1656 375 y Fn(:)0 501 y Fr(By)16 b(explicit)e(in)o(tegration,)h(w)o(e)h(see)g (from)f(this)h(estimate)e(that)244 535 y Fi(\014)244 560 y(\014)244 585 y(\014)244 610 y(\014)274 539 y(Z)315 552 y Fj(t)297 633 y Fm(0)347 598 y Fn(ds)412 539 y Fi(Z)453 552 y Fj(s)435 633 y Fm(0)496 598 y Fk(k)p Fn(V)560 577 y Fm(\(2\))607 598 y Fr(\()p Fn(a)652 577 y Ff(C)675 598 y Fr(\()p Fn(r)o(;)8 b(\017)p Fr(\)\))p Fk(k)15 b Fn(dr)900 535 y Fi(\014)900 560 y(\014)900 585 y(\014)900 610 y(\014)163 746 y Fk(\024)42 b Fn(C)279 753 y Fm(4)307 746 y Fn(t)325 726 y Fm(2)363 746 y Fr(+)20 b Fn(C)456 753 y Fm(5)489 655 y Fi(q)p 530 655 240 2 v 530 705 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))613 691 y Fm(2)633 705 y Fn(t)651 691 y Fm(2)681 705 y Fr(+)11 b Fn(\017)750 691 y Fm(2)780 705 y Fk(\000)g Fn(\017)p 489 735 362 2 v 618 780 a(\032)p Fr(\()p Fn(\017)p Fr(\))701 766 y Fm(2)1825 746 y Fr(\(2.49\))260 875 y(+)16 b Fn(C)349 882 y Fm(6)421 842 y Fr(1)p 382 864 103 2 v 382 909 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))465 895 y Fm(2)506 813 y Fi(\014)506 838 y(\014)506 863 y(\014)506 888 y(\014)528 875 y Fn(\017)11 b Fk(\000)609 823 y Fi(q)p 650 823 240 2 v 650 875 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))733 861 y Fm(2)753 875 y Fn(t)771 861 y Fm(2)801 875 y Fr(+)g Fn(\017)870 861 y Fm(2)901 875 y Fr(+)g Fn(\032)p Fr(\()p Fn(\017)p Fr(\))p Fk(j)p Fn(t)p Fk(j)1087 815 y Fi(\022)1125 875 y Fr(ln)d(\()p Fn(\032)p Fr(\()p Fn(\017)p Fr(\))p Fk(j)p Fn(t)p Fk(j)i Fr(+)1381 823 y Fi(q)p 1423 823 V 52 x Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1506 861 y Fm(2)1525 875 y Fn(t)1543 861 y Fm(2)1573 875 y Fr(+)h Fn(\017)1642 861 y Fm(2)1662 875 y Fr(\))19 b Fk(\000)g Fr(ln)e Fn(\017)1844 815 y Fi(\023)1874 813 y(\014)1874 838 y(\014)1874 863 y(\014)1874 888 y(\014)1896 875 y Fn(:)0 981 y Fr(F)l(or)f(su\016cien)o (tly)f(small)f Fn(T)7 b Fr(,)15 b Fk(j)8 b Fn(t)17 b Fr(ln)f Fn(t)8 b Fk(j)14 b(\024)f Fn(T)7 b Fr(,)15 b Fk(j)8 b Fn(t)17 b Fr(ln)f Fn(\017)8 b Fk(j)14 b(\024)f Fn(T)7 b Fr(,)15 b(and)i Fn(\017)d Fk(\024)f Fn(T)7 b Fr(,)15 b(this)i(quan)o(tit)o(y)e(is)h(b)q(ounded)h(b)o(y)244 1018 y Fi(\014)244 1043 y(\014)244 1068 y(\014)244 1093 y(\014)274 1022 y(Z)315 1035 y Fj(t)297 1116 y Fm(0)347 1080 y Fn(ds)412 1022 y Fi(Z)453 1035 y Fj(s)435 1116 y Fm(0)496 1080 y Fk(k)p Fn(V)560 1060 y Fm(\(2\))607 1080 y Fr(\()p Fn(a)652 1060 y Ff(C)675 1080 y Fr(\()p Fn(r)o(;)8 b(\017)p Fr(\)\))p Fk(k)15 b Fn(dr)900 1018 y Fi(\014)900 1043 y(\014)900 1068 y(\014)900 1093 y(\014)163 1171 y Fk(\024)42 b Fn(C)279 1178 y Fm(7)307 1171 y Fk(j)p Fn(t)p Fk(j)18 b Fr(+)i Fn(C)464 1178 y Fm(8)492 1171 y Fk(j)p Fn(t)c Fr(ln)g Fk(j)p Fn(t)p Fk(j)8 b(j)19 b Fr(+)g Fn(C)776 1178 y Fm(9)804 1171 y Fk(j)p Fn(t)d Fr(ln)g Fn(\017)8 b Fk(j)874 b Fr(\(2.50\))163 1244 y Fk(\024)50 b(O)9 b Fr(\()g Fn(T)14 b Fr(\))9 b Fn(:)0 1321 y Fr(It)17 b(no)o(w)h(follo)o(ws)g(from)e(\(2.47\))j(that)f(for)g (su\016cien)o(tly)d(small)h Fn(T)7 b Fr(,)17 b Fk(jjj)p Fn(A)1281 1303 y Ff(C)1303 1321 y Fr(\()p Fk(\001)p Fn(;)8 b Fk(\001)p Fr(\))j Fk(\000)h Fn(A)1490 1328 y Fm(0)1509 1321 y Fk(jjj)18 b Fr(is)f(b)q(ounded.)26 b(This,)0 1382 y(\(2.47\),)16 b(and)h(\(2.50\))g(imply)d(conclusion)i(\(2.41\))h(of)g (the)f(prop)q(osition.)73 1442 y(T)l(o)h(pro)o(v)o(e)e(\(2.42\),)i(w)o (e)f(use)122 1541 y Fn(B)162 1521 y Ff(C)184 1541 y Fr(\()p Fn(t)p Fr(\))41 b(=)h Fn(B)398 1548 y Fm(0)437 1541 y Fr(+)19 b Fn(i)527 1483 y Fi(Z)569 1496 y Fj(t)550 1577 y Fm(0)608 1541 y Fn(V)647 1521 y Fm(\(2\))695 1541 y Fr(\()p Fn(a)740 1521 y Ff(C)762 1541 y Fr(\()p Fn(s;)8 b(\017)p Fr(\)\))g Fn(A)929 1521 y Ff(C)950 1541 y Fr(\()p Fn(s)p Fr(\))17 b Fn(ds)749 b Fr(\(2.51\))281 1660 y(=)42 b Fn(B)398 1667 y Fm(0)437 1660 y Fr(+)19 b Fn(i)527 1601 y Fi(Z)569 1614 y Fj(t)550 1695 y Fm(0)608 1660 y Fn(V)647 1639 y Fm(\(2\))695 1660 y Fr(\()p Fn(a)740 1639 y Ff(C)762 1660 y Fr(\()p Fn(s;)8 b(\017)p Fr(\)\))g Fn(A)929 1667 y Fm(0)956 1660 y Fn(ds)20 b Fr(+)f Fn(i)1114 1601 y Fi(Z)1156 1614 y Fj(t)1137 1695 y Fm(0)1195 1660 y Fn(V)1234 1639 y Fm(\(2\))1282 1660 y Fr(\()p Fn(a)1327 1639 y Ff(C)1349 1660 y Fr(\()p Fn(s;)8 b(\017)p Fr(\)\))g(\()p Fn(A)1535 1639 y Ff(C)1556 1660 y Fr(\()p Fn(s)p Fr(\))j Fk(\000)g Fn(A)1715 1667 y Fm(0)1735 1660 y Fr(\))d Fn(ds)0 1754 y Fr(F)l(rom)15 b(\(2.45\))i(and)g(explicit)d(in)o(tegration,)243 1795 y Fi(Z)284 1808 y Fj(t)266 1890 y Fm(0)324 1854 y Fn(V)363 1833 y Fm(\(2\))410 1854 y Fr(\()p Fn(a)455 1833 y Ff(C)477 1854 y Fr(\()p Fn(s;)8 b(\017)p Fr(\)\))g Fn(A)644 1861 y Fm(0)672 1854 y Fn(ds)1121 b Fr(\(2.52\))163 1993 y(=)42 b(sign\()p Fn(t)p Fr(\))391 1908 y Fi(0)391 1982 y(@)550 1960 y Fk(j)p Fn(t)p Fk(j)p 432 1982 281 2 v 432 1990 a Fi(q)p 474 1990 240 2 v 50 x Fn(\032)p Fr(\()p Fn(\017)p Fr(\))557 2026 y Fm(2)576 2040 y Fn(t)594 2026 y Fm(2)625 2040 y Fr(+)11 b Fn(\017)694 2026 y Fm(2)726 1993 y Fn(M)5 b(A)815 2000 y Fm(0)854 1993 y Fr(+)946 1960 y(1)p 917 1982 83 2 v 917 2027 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1013 1993 y(ln)1070 1908 y Fi(0)1070 1982 y(@)1111 1960 y Fn(\032)p Fr(\()p Fn(\017)p Fr(\))p Fn(t)p 1111 1982 101 2 v 1152 2027 a(\017)1228 1993 y Fr(+)1277 1910 y Fi(s)p 1318 1910 235 2 v 1323 1960 a Fn(\032)p Fr(\()p Fn(\017)p Fr(\))1406 1945 y Fm(2)1426 1960 y Fn(t)1444 1945 y Fm(2)p 1323 1982 140 2 v 1374 2027 a Fn(\017)1394 2013 y Fm(2)1479 1993 y Fr(+)11 b(1)1553 1908 y Fi(1)1553 1982 y(A)1606 1993 y Fn(N)5 b(A)1687 2000 y Fm(0)1714 1908 y Fi(1)1714 1982 y(A)1770 1993 y Fr(+)19 b Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(:)0 2122 y Fr(F)l(urthermore,)14 b(from)h(\(2.47\),)i(\(2.48\),)f (and\(2.50\),)244 2159 y Fi(\015)244 2184 y(\015)244 2209 y(\015)244 2234 y(\015)275 2163 y(Z)317 2176 y Fj(t)298 2257 y Fm(0)356 2221 y Fn(V)395 2201 y Fm(\(2\))442 2221 y Fr(\()p Fn(a)487 2201 y Ff(C)509 2221 y Fr(\()p Fn(s;)8 b(\017)p Fr(\)\))g(\()p Fn(A)695 2201 y Ff(C)717 2221 y Fr(\()p Fn(s)p Fr(\))j Fk(\000)g Fn(A)876 2228 y Fm(0)895 2221 y Fr(\))d Fn(ds)978 2159 y Fi(\015)979 2184 y(\015)979 2209 y(\015)979 2234 y(\015)1825 2221 y Fr(\(2.53\))163 2341 y Fk(\024)244 2283 y Fi(Z)285 2296 y Fj(t)267 2377 y Fm(0)325 2341 y Fk(k)p Fn(V)389 2321 y Fm(\(2\))436 2341 y Fr(\()p Fn(a)481 2321 y Ff(C)503 2341 y Fr(\()p Fn(s;)g(\017)p Fr(\)\))p Fk(k)g(k)p Fn(A)720 2321 y Ff(C)742 2341 y Fr(\()p Fn(s)p Fr(\))j Fk(\000)f Fn(A)900 2348 y Fm(0)920 2341 y Fk(k)e Fn(ds)824 b Fr(\(2.54\))163 2431 y Fk(\024)42 b Fr([)8 b Fn(C)301 2438 y Fm(1)320 2431 y Fk(j)p Fn(t)p Fk(j)j Fr(+)g Fn(C)461 2438 y Fm(2)491 2431 y Fr(+)g Fk(O)q Fr(\(ln)16 b Fk(j)p Fn(t)p Fk(j)11 b Fr(+)g(ln)16 b Fn(\017)p Fr(\))8 b(])16 b([)8 b Fn(C)954 2438 y Fm(7)974 2431 y Fk(j)p Fn(t)p Fk(j)i Fr(+)h Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)16 b Fr(ln)g Fk(j)p Fn(t)p Fk(j)11 b Fr(+)g Fk(j)p Fn(t)p Fk(j)k Fr(ln)i Fn(\017)p Fr(\))8 b(])g Fn(:)259 b Fr(\(2.55\))0 2508 y(F)l(or)16 b Fn(t)g Fr(and)h Fn(\017)f Fr(in)g(the)g(regime)e(co)o(v)o(ered)h(b)o (y)h(the)g(prop)q(osition,)h(this)f(last)g(quan)o(tit)o(y)f(is)h Fk(O)q Fr(\()8 b Fn(t)17 b Fr(ln)1721 2487 y Fm(2)1758 2508 y Fn(\017)8 b Fr(\).)73 2568 y(Conclusion)17 b(\(2.42\))g(is)f(no) o(w)g(an)h(easy)f(consequence)g(of)g(\(2.51\),)h(\(2.52\),)f(and)h (\(2.53\).)251 b Fa(2)951 2753 y Fr(22)p eop %%Page: 23 23 23 22 bop 0 -22 a Fc(2.7)70 b(Alterations)21 b(for)j(Other)e(T)n(yp)r (es)h(of)g(Av)n(oided)g(Crossings)0 71 y Fr(The)16 b(analysis)h(presen) o(ted)e(ab)q(o)o(v)o(e)h(is)g(iden)o(tical)f(for)h(T)o(yp)q(e)g(4)h(Av) o(oided)e(Crossings.)73 161 y(F)l(or)j(T)o(yp)q(e)f(5)i(Av)o(oided)d (Crossings,)j(one)f(m)o(ust)e(mak)o(e)g(the)i(follo)o(wing)f (substitutions)h(in)g(the)f(ab)q(o)o(v)o(e)0 221 y(results,)f(but)g (the)g(tec)o(hniques)f(of)h(pro)q(of)i(remain)c(the)i(same.)72 468 y Fn(b)e Fr(=)158 283 y Fi(0)158 356 y(B)158 381 y(B)158 406 y(B)158 430 y(B)158 455 y(B)158 480 y(B)158 505 y(B)158 530 y(B)158 557 y(@)203 314 y Fn(b)224 321 y Fm(1)203 374 y Fn(b)224 381 y Fm(2)203 434 y Fn(b)224 441 y Fm(3)211 494 y Fr(0)217 528 y(.)217 545 y(.)217 562 y(.)211 622 y(0)252 283 y Fi(1)252 356 y(C)252 381 y(C)252 406 y(C)252 430 y(C)252 455 y(C)252 480 y(C)252 505 y(C)252 530 y(C)252 557 y(A)296 468 y Fn(;)57 b(\032)p Fr(\()p Fn(\017)p Fr(\))14 b(=)520 434 y Fn(b)541 441 y Fm(1)561 434 y Fn(\021)587 416 y Fm(0)585 446 y(1)607 434 y Fr(\()p Fn(\017)p Fr(\))p 520 456 144 2 v 557 502 a Fk(j)p Fn(e)594 509 y Fm(4)613 502 y Fk(j)669 468 y Fn(;)57 b(M)19 b Fr(=)886 434 y(1)p 863 456 70 2 v 863 502 a Fk(j)p Fn(e)900 509 y Fm(4)919 502 y Fk(j)946 468 y(j)p Fn(b)p Fk(ih)p Fn(b)p Fk(j)p Fn(;)57 b Fr(and)49 b Fn(N)19 b Fr(=)1390 434 y(1)p 1367 456 V 1367 502 a Fk(j)p Fn(e)1404 509 y Fm(4)1423 502 y Fk(j)1458 468 y Fr(\()9 b Fk(j)p Fn(c)p Fk(ih)p Fn(c)p Fk(j)19 b Fr(+)h Fk(j)p Fn(d)p Fk(ih)p Fn(d)p Fk(j)8 b Fr(\))h Fn(:)73 744 y Fr(Similarly)l(,)j(for)k(T)o(yp)q(e)f(6)h(Av)o(oided)f (Crossings,)h(one)g(m)o(ust)e(mak)o(e)g(the)h(follo)o(wing)g (substitutions,)h(but)0 805 y(the)g(tec)o(hniques)f(of)h(pro)q(of)i (remain)c(the)i(same.)31 1081 y Fn(b)d Fr(=)117 871 y Fi(0)117 944 y(B)117 969 y(B)117 994 y(B)117 1019 y(B)117 1044 y(B)117 1069 y(B)117 1094 y(B)117 1119 y(B)117 1144 y(B)117 1169 y(B)117 1195 y(@)162 897 y Fn(b)183 904 y Fm(1)162 958 y Fn(b)183 965 y Fm(2)162 1018 y Fn(b)183 1025 y Fm(3)162 1078 y Fn(b)183 1085 y Fm(4)170 1138 y Fr(0)175 1172 y(.)175 1189 y(.)175 1205 y(.)170 1265 y(0)210 871 y Fi(1)210 944 y(C)210 969 y(C)210 994 y(C)210 1019 y(C)210 1044 y(C)210 1069 y(C)210 1094 y(C)210 1119 y(C)210 1144 y(C)210 1169 y(C)210 1195 y(A)255 1081 y Fn(;)24 b(\032)p Fr(\()p Fn(\017)p Fr(\))14 b(=)446 1048 y Fn(b)467 1055 y Fm(1)487 1048 y Fn(\021)513 1030 y Fm(0)511 1060 y(1)532 1048 y Fr(\()p Fn(\017)p Fr(\))p 446 1070 144 2 v 483 1116 a Fk(j)p Fn(f)521 1123 y Fm(5)540 1116 y Fk(j)595 1081 y Fn(;)24 b(M)19 b Fr(=)780 1048 y(1)p 756 1070 72 2 v 756 1116 a Fk(j)p Fn(f)794 1123 y Fm(5)814 1116 y Fk(j)841 1081 y(j)p Fn(b)p Fk(ih)p Fn(b)p Fk(j)p Fn(;)24 b Fr(and)17 b Fn(N)i Fr(=)1220 1048 y(1)p 1197 1070 V 1197 1116 a Fk(j)p Fn(f)1235 1123 y Fm(5)1254 1116 y Fk(j)1290 1081 y Fr(\()8 b Fk(j)p Fn(c)p Fk(ih)p Fn(c)p Fk(j)20 b Fr(+)f Fk(j)p Fn(d)p Fk(ih)p Fn(d)p Fk(j)h Fr(+)g Fk(j)p Fn(e)p Fk(ih)p Fn(e)p Fk(j)8 b Fr(\))g Fn(:)0 1448 y Fo(3)83 b(Propagation)28 b(through)g(T)n(yp)r(e)f(3)g(Av)n(oided)g(Crossings)0 1558 y Fr(W)l(e)16 b(no)o(w)h(ha)o(v)o(e)e(all)h(the)g(ingredien)o(ts)f (to)i(construct)f(an)h(asymptotic)e(solution)h(to)h(the)f(equation)122 1686 y Fn(i\017)159 1666 y Fm(2)192 1653 y Fn(@)p 183 1675 47 2 v 183 1721 a(@)s(t)234 1686 y( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))k(=)i Fk(\000)482 1653 y Fn(\017)502 1635 y Fm(4)p 482 1675 40 2 v 489 1721 a Fr(2)526 1686 y(\001)p Fn( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))i(+)h Fn(h)p Fr(\()p Fn(x;)d(\017)p Fr(\))p Fn( )r Fr(\()p Fn(x;)g(t)p Fr(\))808 b(\(3.1\))0 1801 y(as)17 b Fn(\017)c Fk(!)h Fr(0.)22 b(W)l(e)16 b(\014rst)g(presen)o(t)g(the)g(building)g(blo)q(c)o(ks)f (and)i(giv)o(e)f(their)f(main)g(prop)q(erties.)0 1945 y Fc(3.1)70 b(Semicl)o(assical)19 b(Nuclear)j(W)-6 b(a)n(v)n(e)23 b(P)n(ac)n(k)n(ets)0 2038 y Fr(The)15 b(semiclassical)e(motion)h(of)h (the)g(n)o(uclei)e(is)i(describ)q(ed)g(b)o(y)f(w)o(a)o(v)o(e)g(pac)o(k) o(ets)g(whic)o(h)h(corresp)q(ond)h(to)f(the)0 2098 y(classical)i(phase) i(space)f(tra)s(jectory)l(,)f(and)h(of)h(width)e Fk(O)q Fr(\()p Fn(\017)p Fr(\).)26 b(These)18 b(are)g(the)g(same)f(w)o(a)o(v)o (e)g(pac)o(k)o(ets)g(that)0 2158 y(are)f(used)h(in)f([8,)f(17)q(,)h(20) q(].)73 2218 y(Let)25 b Fn(n)h Fr(denote)f(the)f(dimension)g(of)h(the)g (n)o(uclear)f(con\014guration)i(space.)48 b(A)25 b(m)o(ulti-index)d Fn(l)29 b Fr(=)0 2278 y(\()p Fn(l)34 2285 y Fm(1)53 2278 y Fn(;)16 b(l)98 2285 y Fm(2)118 2278 y Fn(;)g(:)8 b(:)g(:)f(;)16 b(l)258 2285 y Fj(n)281 2278 y Fr(\))24 b(is)f(an)g(ordered)g Fn(n)p Fr(-tuple)g(of)h(non{negativ)o(e)f(in)o(tegers.)41 b(The)24 b(order)f(of)g Fn(l)h Fr(is)f(de\014ned)0 2339 y(to)17 b(b)q(e)g Fk(j)p Fn(l)q Fk(j)d Fr(=)238 2305 y Fi(P)281 2319 y Fj(n)281 2351 y(j)r Fm(=1)361 2339 y Fn(l)376 2346 y Fj(j)394 2339 y Fr(,)j(and)g(the)g(factorial)f(of)h Fn(l)h Fr(is)e(de\014ned)h(to)g(b)q(e)g Fn(l)q Fr(!)c(=)i(\()p Fn(l)1360 2346 y Fm(1)1379 2339 y Fr(!\)\()p Fn(l)1446 2346 y Fm(2)1465 2339 y Fr(!\))8 b Fk(\001)g(\001)g(\001)g Fr(\()p Fn(l)1606 2346 y Fj(n)1629 2339 y Fr(!\).)23 b(The)16 b(sym)o(b)q(ol)0 2412 y Fn(D)41 2393 y Fj(l)70 2412 y Fr(denotes)g(the)f(di\013eren)o(tial)f(op)q(erator)i Fn(D)808 2393 y Fj(l)836 2412 y Fr(=)1057 2392 y Fj(@)1078 2380 y Fe(j)p Fh(l)p Fe(j)p 893 2400 381 2 v 893 2430 a Fm(\()p Fj(@)r(x)948 2435 y Fd(1)964 2430 y Fm(\))978 2419 y Fh(l)988 2426 y Fd(1)1007 2430 y Fm(\()p Fj(@)r(x)1062 2435 y Fd(2)1079 2430 y Fm(\))1093 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Fm(2)467 2556 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)123 2629 y Fr(\010)158 2608 y Ff(\000)158 2641 y(B)187 2629 y Fr(\()p Fn(x;)g(\017)p Fr(\))41 b(=)h Fk(\000)p Fn( )487 2636 y Fm(1)506 2629 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)951 2753 y Fr(26)p eop %%Page: 27 27 27 26 bop 0 -22 a Fl(and)18 b(for)e Fn(t)e(>)g Fr(0)p Fl(,)122 80 y Fr(\010)157 60 y Fm(+)157 92 y Ff(A)187 80 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)h Fn( )448 87 y Fm(1)467 80 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)123 153 y Fr(\010)158 132 y Fm(+)158 165 y Ff(B)187 153 y Fr(\()p Fn(x;)g(\017)p Fr(\))41 b(=)h Fn( )448 160 y Fm(2)467 153 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(:)0 267 y Fg(Pro)r(of:)0 327 y Fr(Let)16 b Fn(x)g Fr(b)q(elong)h(to)g(the)f(supp)q(ort)h(of)g Fn(F)7 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Fk(O)q Fr(\()p Fn(\017)555 614 y Fm(1)p Ff(\000)p Fj(\016)617 602 y Fe(0)630 634 y Fr(\))p Fn(;)105 b Fr(for)16 b Fn(j)h Fk(\025)d Fr(3)p Fn(:)0 736 y Fr(Th)o(us,)122 838 y Fn(x)150 845 y Fm(1)211 838 y Fr(=)41 b Fn(\021)316 817 y Fm(0)314 850 y(1)336 838 y Fn(t)11 b Fr(+)g Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)122 910 y(x)150 917 y Fm(2)211 910 y Fr(=)41 b Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\)\))p Fn(;)123 983 y(x)151 990 y Fj(j)211 983 y Fr(=)41 b Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(;)105 b Fr(for)17 b Fn(j)f Fk(\025)e Fr(3)p Fn(:)0 1085 y Fr(Therefore,)122 1186 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(b)401 1193 y Fm(1)421 1186 y Fn(\021)447 1166 y Fm(0)445 1199 y(1)466 1186 y Fn(t)11 b Fr(+)g Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)1015 b Fr(\(3.27\))124 1259 y Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\)\))p Fn(;)1179 b Fr(\(3.28\))129 1332 y Fn(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(d)405 1339 y Fm(3)425 1332 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))13 b(=)h Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\)\))p Fn(:)737 b Fr(\(3.29\))0 1433 y(F)l(rom)15 b(these)h(estimates,)e(it)i(follo)o(ws)g(that)127 1541 y Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))p 127 1564 138 2 v 184 1609 a Fn(r)311 1575 y Fr(=)443 1541 y(sign\()p Fn(t)p Fr(\))p 395 1564 235 2 v 395 1572 a Fi(q)p 437 1572 194 2 v 49 x Fr(1)j(+)526 1600 y Fj(\015)546 1590 y Fd(2)564 1600 y Fm(+)p Fj(\016)608 1590 y Fd(2)p 526 1609 100 2 v 556 1638 a Fj(\014)578 1629 y Fd(2)649 1575 y Fr(=)j(sign\()p Fn(t)p Fr(\))865 1502 y Fi( )906 1575 y Fr(1)e(+)996 1541 y Fk(O)d Fr(\(\()p Fk(j)p Fn(t)p Fk(j)p Fn(\026)p Fr(\()p Fn(\017;)f(t)p Fr(\)\))1275 1523 y Fm(2)1294 1541 y Fr(\))p 996 1564 317 2 v 1096 1609 a Fk(O)q Fr(\()p Fn(t)1174 1595 y Fm(2)1193 1609 y Fr(\))1318 1502 y Fi(!)1350 1514 y Ff(\000)p Fm(1)p Fj(=)p Fm(2)1447 1575 y Fr(=)13 b(sign)q(\()p Fn(t)p Fr(\))d(+)h Fk(O)1748 1527 y Fi(\020)1772 1575 y Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\))1899 1555 y Fm(2)1919 1527 y Fi(\021)1952 1575 y Fn(;)129 1699 y(\015)s Fr(\()p Fn(x;)g(\017)p Fr(\))p 129 1721 136 2 v 185 1767 a Fn(r)311 1733 y Fr(=)41 b Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\)\))p Fn(;)134 1821 y(\016)r Fr(\()p Fn(x;)g(\017)p Fr(\))p 134 1843 131 2 v 187 1889 a Fn(r)311 1854 y Fr(=)41 b Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\)\))p Fn(:)0 1976 y Fr(Since)32 b(sin)203 1956 y Fm(2)223 1976 y Fr(\()p Fn(\022)q(=)p Fr(2\))15 b(=)405 1956 y Fm(1)p 405 1964 18 2 v 405 1993 a(2)427 1976 y Fr(\(1)d Fk(\000)e Fr(cos)f(\()p Fn(\022)q Fr(\)\),)16 b(and)33 b(cos)892 1958 y Fm(2)912 1976 y Fr(\()p Fn(\022)q(=)p Fr(2\))15 b(=)e(1)f Fk(\000)f Fr(sin)1234 1956 y Fm(2)1253 1976 y Fr(\()p Fn(\022)q(=)p Fr(2\),)17 b(w)o(e)f(see)g(that)130 2106 y(sin)190 2085 y Fm(2)210 2106 y Fr(\()p Fn(\022)q(=)p Fr(2\))42 b(=)442 2045 y Fi(\032)481 2076 y Fr(0)49 b(if)16 b Fn(t)d(>)h Fr(0)481 2136 y(1)49 b(if)16 b Fn(t)d(<)h Fr(0)715 2045 y Fi(\033)765 2106 y Fr(+)20 b Fk(O)872 2058 y Fi(\020)897 2106 y Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\))1024 2085 y Fm(2)1043 2058 y Fi(\021)1271 2106 y Fr(and)476 b(\(3.30\))125 2232 y(cos)190 2212 y Fm(2)210 2232 y Fr(\()p Fn(\022)q(=)p Fr(2\))42 b(=)442 2172 y Fi(\032)481 2202 y Fr(1)49 b(if)16 b Fn(t)d(>)h Fr(0)481 2263 y(0)49 b(if)16 b Fn(t)d(<)h Fr(0)715 2172 y Fi(\033)765 2232 y Fr(+)20 b Fk(O)872 2184 y Fi(\020)897 2232 y Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\))1024 2212 y Fm(2)1043 2184 y Fi(\021)1076 2232 y Fn(:)0 2355 y Fr(The)16 b(lemma)d(follo)o(ws.)1498 b Fa(2)73 2475 y Fr(W)l(e)16 b(no)o(w)h(construct)f(the)g(dynamic)f(eigen)o(v)o (ectors.)951 2753 y(27)p eop %%Page: 28 28 28 27 bop 0 -22 a Fg(Lemma)16 b(3.2)24 b Fl(Supp)n(ose)c Fr(0)f Fn(<)g(\016)607 -40 y Ff(0)636 -22 y Fn(<)g(\030)i(<)797 -41 y Fm(1)p 797 -33 18 2 v 797 -4 a(3)820 -22 y Fl(,)f Fk(j)p Fn(t)p Fk(j)e Fn(>)h(\017)996 -40 y Fm(1)p Ff(\000)p Fj(\030)1060 -22 y Fl(,)h(and)h Fk(j)p Fn(t)p Fk(j)d Fn(<)g(\017)1333 -40 y Fj(\024)1376 -22 y Fl(for)h(some)h Fn(\024)f(>)1692 -41 y Fm(1)p 1692 -33 V 1692 -4 a(2)1714 -22 y Fl(.)31 b(L)n(et)19 b Fn(a)1872 -40 y Ff(C)1894 -22 y Fr(\()p Fn(t)p Fr(\))0 39 y Fl(and)h Fn(\021)123 21 y Ff(C)146 39 y Fr(\()p Fn(t)p Fr(\))f Fl(solve)i(the)g(classic)n (al)g(e)n(quations)g(of)f(motion)g(\(2.2\))f(and)i(\(2.3\).)29 b(Ther)n(e)20 b(exist)h(eigenve)n(ctors)0 99 y Fr(\010)35 78 y Ff(\006)35 111 y(C)65 99 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))p Fl(,)16 b Fk(C)h Fr(=)d Fk(A)p Fn(;)8 b Fk(B)r Fl(,)16 b(that)h(solve)i(\(3.19\))e(and)h(satisfy)644 162 y Fi(\014)644 187 y(\014)644 212 y(\014)p Fr(\010)693 191 y Ff(\006)693 224 y(C)723 212 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))h Fk(\000)i Fr(\010)965 191 y Ff(\006)965 224 y(C)995 212 y Fr(\()p Fn(x;)d(\017)p Fr(\))1103 162 y Fi(\014)1103 187 y(\014)1103 212 y(\014)13 b Fr(=)h Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))0 331 y Fl(for)j(al)r(l)i Fn(x)e Fl(in)h(the)f(supp)n(ort)g(of)g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)j Fk(\000)h Fn(a)760 313 y Ff(C)782 331 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)907 313 y Fm(1)p Ff(\000)p Fj(\016)969 301 y Fe(0)982 331 y Fr(\))p Fl(.)0 441 y Fg(Pro)r(of:)0 501 y Fr(W)l(e)20 b(giv)o(e)g(a)h(pro)q(of)g(for)g (\010)483 481 y Fm(+)483 514 y Ff(A)514 501 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))19 b(only;)j(the)e(other)h(cases)g(are)g (similar.)32 b(In)20 b(order)g(to)h(simplify)d(the)0 562 y(notation,)e(w)o(e)g(drop)h(the)e(indices)g Fk(A)h Fr(and)h(+,)e(and)i(let)e Fn(a)p Fr(\()p Fn(t)p Fr(\))e Fk(\021)h Fn(a)1194 543 y Ff(A)1224 562 y Fr(\()p Fn(t)p Fr(\))h(and)i Fn(\021)r Fr(\()p Fn(t)p Fr(\))c Fk(\021)h Fn(\021)1564 543 y Ff(A)1594 562 y Fr(\()p Fn(t)p Fr(\))h(for)i Fn(t)c(>)h Fr(0.)21 b(W)l(e)0 622 y(again)c(let)f Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\))13 b(=)h(max)7 b Fk(f)h(j)p Fn(t)p Fk(j)p Fn(;)15 b(\017)620 604 y Fm(1)p Ff(\000)p Fj(\016)682 592 y Fe(0)695 622 y Fn(=)p Fk(j)p Fn(t)p Fk(j)8 b(g)p Fr(.)73 682 y(Solutions)17 b(of)f(\(3.19\))h(ha)o(v)o(e)f (the)g(form)122 781 y(\010\()p Fn(x;)8 b(t;)g(\017)p Fr(\))13 b(=)g(\010\()p Fn(x;)8 b(\017)p Fr(\)e)534 760 y Fj(i\025)p Fm(\()p Fj(x;t;\017)p Fm(\))662 781 y Fn(;)1149 b Fr(\(3.31\))0 880 y(where)16 b Fn(\025)p Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))15 b(is)h(a)h(real)f(v)m(alued)g(function)g(that)h (satis\014es)f(the)g(equation)122 1003 y Fn(i)144 969 y(@)p 144 991 47 2 v 144 1037 a(@)s(t)195 1003 y(\025)p Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))i(+)h Fn(i\021)r Fr(\()p Fn(t)p Fr(\))f Fk(\001)h(r)p Fn(\025)p Fr(\()p Fn(x;)d(t;)g(\017)p Fr(\))i(+)h Fk(h)p Fr(\010\()p Fn(x;)d(\017)p Fr(\))p Fn(;)15 b(\021)r Fr(\()p Fn(t)p Fr(\))c Fk(\001)g(r)p Fr(\010\()p Fn(x;)d(\017)p Fr(\))p Fk(i)13 b Fr(=)h(0)p Fn(:)368 b Fr(\(3.32\))0 1115 y(W)l(e)16 b(in)o(tro)q(duce)122 1213 y Fn(!)g Fk(\021)e Fn(x)c Fk(\000)h Fn(a)p Fr(\()p Fn(t)p Fr(\))1434 b(\(3.33\))0 1312 y(and)17 b(de\014ne)208 1394 y Fi(e)205 1411 y Fn(\025)p Fr(\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b Fk(\021)h Fn(\025)p Fr(\()p Fn(!)g Fr(+)d Fn(a)p Fr(\()p Fn(t)p Fr(\))p Fn(;)d(t;)g(\017)p Fr(\))p Fn(;)1037 b Fr(\(3.34\))211 1468 y Fi(e)205 1484 y Fr(\010\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b Fk(\021)h Fr(\010\()p Fn(!)f Fr(+)e Fn(a)p Fr(\()p Fn(t)p Fr(\))p Fn(;)d(\017)p Fr(\))p Fn(:)1064 b Fr(\(3.35\))0 1583 y(In)16 b(the)g(new)g(v)m (ariables,)g(equation)g(\(3.32\))h(for)878 1566 y Fi(e)875 1583 y Fn(\025)g Fr(is)122 1706 y Fn(i)144 1672 y(@)p 144 1695 V 144 1740 a(@)s(t)197 1689 y Fi(e)195 1706 y Fn(\025)p Fr(\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b(=)h Fk(\000h)504 1690 y Fi(e)498 1706 y Fr(\010\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))p Fn(;)728 1672 y(@)p 719 1695 V 719 1740 a(@)s(t)776 1690 y Fi(e)770 1706 y Fr(\010)q(\()p Fn(!)r(;)g(t;)g(\017)p Fr(\))p Fk(i)848 b Fr(\(3.36\))0 1818 y(where)127 1895 y Fn(@)p 127 1917 V 127 1963 a(@)s(t)184 1913 y Fi(e)178 1929 y Fr(\010\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b(=)h Fn(\021)r Fr(\()p Fn(t)p Fr(\))c Fk(\001)h(r)p Fr(\010\()p Fn(!)i Fr(+)e Fn(a)p Fr(\()p Fn(t)p Fr(\))p Fn(;)d(\017)p Fr(\))p Fn(:)933 b Fr(\(3.37\))0 2040 y(Dropping)17 b(the)f(argumen)o(ts,)f(w)o(e)h(ha)o (v)o(e)122 2139 y Fk(r)p Fr(\010)e(=)333 2091 y Fi(\020)358 2139 y Fk(\000)8 b Fr(sin\()p Fn(\022)q(=)p Fr(2\))p Fn( )607 2146 y Fm(1)638 2139 y Fr(+)j(e)709 2119 y Ff(\000)p Fj(i')782 2139 y Fr(cos\()p Fn(\022)q(=)p Fr(2\))p Fn( )989 2146 y Fm(2)1010 2091 y Fi(\021)1043 2139 y Fk(r)p Fn(\022)q(=)p Fr(2)333 2223 y(+)d(cos)q(\()p Fn(\022)q(=)p Fr(2\))p Fk(r)p Fn( )629 2230 y Fm(1)660 2223 y Fr(+)j(e)730 2203 y Ff(\000)p Fj(i')803 2223 y Fr(sin\()p Fn(\022)q(=)p Fr(2\))p Fk(r)p Fn( )1047 2230 y Fm(2)333 2296 y Fk(\000)p Fn(i)p Fk(r)p Fn(')p Fr(e)484 2275 y Ff(\000)p Fj(i')557 2296 y Fr(sin\()p Fn(\022)q(=)p Fr(2\))p Fn( )759 2303 y Fm(2)779 2296 y Fn(:)1032 b Fr(\(3.38\))73 2395 y(Since)16 b(the)g Fn( )317 2402 y Fj(j)334 2395 y Fr(,)g Fn(j)h Fr(=)d(1)p Fn(;)8 b Fr(2,)16 b(are)h(orthonormal,)e(w)o(e)h(get)122 2520 y Fk(h)147 2504 y Fi(e)141 2520 y Fr(\010)q Fn(;)220 2486 y(@)p 212 2508 V 212 2554 a(@)s(t)269 2504 y Fi(e)263 2520 y Fr(\010)p Fk(i)e Fr(=)42 b Fk(\000)f Fn(i\021)r Fk(r)p Fn(')8 b Fr(sin)675 2499 y Fm(2)695 2520 y Fr(\()p Fn(\022)q(=)p Fr(2\))k(+)f(cos)931 2499 y Fm(2)951 2520 y Fr(\()p Fn(\022)q(=)p Fr(2\))p Fk(h)p Fn( )1112 2527 y Fm(1)1132 2520 y Fk(j)p Fn(\021)r Fk(r)p Fn( )1246 2527 y Fm(1)1265 2520 y Fk(i)g Fr(+)g(sin)1404 2499 y Fm(2)1424 2520 y Fr(\()p Fn(\022)q(=)p Fr(2\))p Fk(h)p Fn( )1585 2527 y Fm(2)1605 2520 y Fk(j)p Fn(\021)r Fk(r)p Fn( )1719 2527 y Fm(2)1738 2520 y Fk(i)411 2620 y Fr(+)42 b(sin\()p Fn(\022)q(=)p Fr(2\))8 b(cos)q(\()p Fn(\022)q(=)p Fr(2\))853 2572 y Fi(\020)879 2620 y Fr(e)901 2600 y Ff(\000)p Fj(i')965 2620 y Fk(h)p Fn( )1016 2627 y Fm(1)1036 2620 y Fk(j)p Fn(\021)r Fk(r)p Fn( )1150 2627 y Fm(2)1169 2620 y Fk(i)j Fr(+)g(e)1270 2600 y Fj(i')1307 2620 y Fk(h)p Fn( )1358 2627 y Fm(2)1378 2620 y Fk(j)p Fn(\021)r Fk(r)p Fn( )1492 2627 y Fm(1)1510 2620 y Fk(i)1529 2572 y Fi(\021)1563 2620 y Fn(:)248 b Fr(\(3.39\))951 2753 y(28)p eop %%Page: 29 29 29 28 bop 0 -22 a Fr(As)16 b Fn(\021)r(;)8 b( )152 -15 y Fj(j)170 -22 y Fn(;)g Fk(r)p Fn( )266 -15 y Fj(j)299 -22 y Fr(are)17 b(all)e Fk(O)q Fr(\(0\),)i(w)o(e)e(ha)o(v)o(e)125 74 y Fi(e)122 91 y Fn(\025)p Fr(\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b(=)367 32 y Fi(Z)409 45 y Fj(t)390 126 y Fm(0)432 42 y Fi(\020)457 91 y Fn(\021)r Fr(\()p Fn(t)520 70 y Ff(0)531 91 y Fr(\))p Fk(r)p Fn(')8 b Fr(sin)691 70 y Fm(2)711 91 y Fr(\()p Fn(\022)q Fr(\()p Fn(!)13 b Fr(+)e Fn(a)p Fr(\()p Fn(t)928 70 y Ff(0)939 91 y Fr(\))p Fn(;)d(\017)p Fr(\))p Fn(=)p Fr(2\))j(+)g Fk(O)q Fr(\(0\))1249 42 y Fi(\021)1283 91 y Fn(dt)1326 70 y Ff(0)1337 91 y Fn(;)474 b Fr(\(3.40\))0 193 y(where)16 b(w)o(e)g(ha)o(v)o(e)f(no)o(w)i(\014xed) f(the)g(constan)o(t)g(of)h(in)o(tegration.)73 254 y(W)l(e)g(claim)e (that)j Fk(j)p Fn(\021)13 b Fk(\001)f(r)p Fn(')k Fr(sin)623 234 y Fm(2)659 254 y Fn(\022)q(=)p Fr(2)p Fk(j)i Fr(is)f(b)q(ounded)h (on)g(the)f(supp)q(ort)h(of)g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)k Fk(\000)g Fn(a)1603 236 y Ff(C)1625 254 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)1750 236 y Fm(1)p Ff(\000)p Fj(\016)1812 224 y Fe(0)1825 254 y Fr(\).)24 b(W)l(e)0 314 y(note)16 b(that)122 402 y Fn(')p Fr(\()p Fn(x;)8 b(\017)p Fr(\))13 b(=)h(cot)392 382 y Ff(\000)p Fm(1)447 402 y Fr(\()p Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))p Fn(=\016)r Fr(\()p Fn(x;)g(\017)p Fr(\)\))p Fn(;)1034 b Fr(\(3.41\))0 491 y(pro)o(vided)14 b Fn(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))13 b(is)i(di\013eren)o(t)f(from)g(zero.)20 b(Ho)o(w)o(ev)o(er,)13 b(on)i(the)g(supp)q(ort)h(of)f Fn(F)7 b Fr(\()p Fk(k)p Fn(x)h Fk(\000)g Fn(a)1601 473 y Ff(C)1622 491 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)1747 473 y Fm(1)p Ff(\000)p Fj(\016)1809 461 y Fe(0)1821 491 y Fr(\))15 b(with)0 551 y Fk(j)p Fn(t)p Fk(j)e Fn(<)h(\017)131 533 y Fj(\024)153 551 y Fr(,)122 640 y Fn(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(d)399 647 y Fm(3)419 640 y Fn(\017)20 b Fr(+)f Fk(O)q Fr(\()p Fn(x)604 619 y Fm(2)634 640 y Fr(+)11 b Fn(\017)703 619 y Fm(2)723 640 y Fr(\))295 712 y(=)41 b Fn(d)399 719 y Fm(3)419 712 y Fn(\017)11 b Fr(+)g Fk(O)q Fr(\()p Fn(t)577 692 y Fm(2)608 712 y Fr(+)g Fn(\017)677 692 y Fm(2)696 712 y Fr(\))294 785 y Fk(\025)41 b Fn(C)t(\017;)202 b Fr(for)17 b(some)e(p)q(ositiv)o(e)h Fn(C)q(:)73 873 y Fr(By)g(the)g(mean)f(v)m (alue)h(theorem,)122 931 y Fi(q)p 163 931 150 2 v 163 984 a Fn(y)189 969 y Fm(2)220 984 y Fr(+)11 b Fn(z)294 969 y Fm(2)355 984 y Fr(=)42 b Fk(j)p Fn(y)r Fk(j)10 b Fr(+)h Fk(j)p Fn(z)r Fk(j)698 950 y Fn(\020)p 614 972 192 2 v 614 981 a Fk(p)p 656 981 151 2 v 38 x Fn(y)682 1005 y Fm(2)712 1019 y Fr(+)g Fn(\020)786 1005 y Fm(2)909 984 y Fr(for)16 b(some)f Fn(\020)j Fk(2)c Fr([)8 b(0)p Fn(;)17 b Fk(j)p Fn(z)r Fk(j)8 b Fr(])355 1083 y Fk(\024)41 b(j)p Fn(y)r Fk(j)10 b Fr(+)h Fk(j)p Fn(z)r Fk(j)p Fn(:)0 1171 y Fr(Th)o(us,)122 1289 y(2)17 b(sin)223 1269 y Fm(2)259 1289 y Fn(\022)q(=)p Fr(2)23 b(=)f(1)12 b Fk(\000)e Fr(cos)17 b Fn(\022)24 b Fr(=)693 1256 y Fn(r)12 b Fk(\000)f Fn(\014)p 693 1278 115 2 v 738 1323 a(r)834 1289 y Fr(=)899 1218 y Fk(p)p 941 1218 262 2 v 38 x Fn(\014)972 1241 y Fm(2)1002 1256 y Fr(+)g Fn(\015)1079 1241 y Fm(2)1110 1256 y Fr(+)g Fn(\016)1183 1241 y Fm(2)1213 1256 y Fk(\000)g Fn(\014)p 899 1278 394 2 v 1085 1323 a(r)0 1390 y Fr(satis\014es)122 1449 y Fi(\014)122 1474 y(\014)122 1499 y(\014)d Fr(2)17 b(sin)245 1478 y Fm(2)281 1499 y Fn(\022)q(=)p Fr(2)361 1449 y Fi(\014)362 1474 y(\014)362 1499 y(\014)42 b Fk(\024)558 1427 y(p)p 600 1427 151 2 v 38 x Fn(\015)628 1450 y Fm(2)658 1465 y Fr(+)11 b Fn(\016)731 1450 y Fm(2)p 503 1487 303 2 v 503 1496 a Fk(p)p 545 1496 262 2 v 38 x Fn(\014)576 1520 y Fm(2)606 1534 y Fr(+)g Fn(\015)683 1520 y Fm(2)713 1534 y Fr(+)g Fn(\016)786 1520 y Fm(2)811 1499 y Fn(:)73 1614 y Fr(F)l(rom)k(\(3.41\))i(w)o(e)f(therefore)f(see)h(that)122 1680 y Fi(\014)122 1705 y(\014)122 1730 y(\014)8 b Fn(\021)13 b Fk(\001)e(r)p Fn(')16 b Fr(sin)356 1709 y Fm(2)392 1730 y Fn(\022)q(=)p Fr(2)472 1680 y Fi(\014)473 1705 y(\014)473 1730 y(\014)42 b Fk(\024)628 1696 y(j)8 b Fn(\015)k(\021)h Fk(\001)d(r)p Fn(\016)g Fk(j)h Fr(+)g Fk(j)d Fn(\016)i(\021)i Fk(\001)f(r)p Fn(\015)g Fk(j)p 614 1718 504 2 v 614 1727 a(p)p 655 1727 151 2 v 655 1765 a Fn(\015)683 1751 y Fm(2)714 1765 y Fr(+)g Fn(\016)787 1751 y Fm(2)815 1727 y Fk(p)p 856 1727 262 2 v 856 1765 a Fn(\014)887 1751 y Fm(2)917 1765 y Fr(+)g Fn(\015)994 1751 y Fm(2)1025 1765 y Fr(+)g Fn(\016)1098 1751 y Fm(2)529 1862 y Fk(\024)614 1829 y(j)d Fn(\021)13 b Fk(\001)e(r)p Fn(\015)g Fk(j)19 b Fr(+)g Fk(j)8 b Fn(\021)13 b Fk(\001)e(r)p Fn(\016)e Fk(j)p 614 1851 424 2 v 674 1860 a(p)p 716 1860 262 2 v 38 x Fn(\014)747 1883 y Fm(2)777 1898 y Fr(+)i Fn(\015)854 1883 y Fm(2)885 1898 y Fr(+)g Fn(\016)958 1883 y Fm(2)0 1984 y Fr(Ho)o(w)o(ev)o(er,)j(on)j(the)f(supp)q(ort)h(of) g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)j Fk(\000)h Fn(a)795 1966 y Ff(C)817 1984 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)942 1966 y Fm(1)p Ff(\000)p Fj(\016)1004 1954 y Fe(0)1016 1984 y Fr(\),)122 2032 y Fi(q)p 163 2032 V 163 2085 a Fn(\014)194 2071 y Fm(2)225 2085 y Fr(+)g Fn(\015)302 2071 y Fm(2)332 2085 y Fr(+)g Fn(\016)405 2071 y Fm(2)466 2085 y Fk(\025)41 b Fn(C)t Fk(j)p Fn(t)p Fk(j)p Fn(;)104 b Fr(for)17 b(some)e Fn(C)q(;)253 2158 y Fk(j)8 b Fn(\021)14 b Fk(\001)c(r)p Fn(\016)g Fk(j)41 b Fr(=)h Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(;)203 b Fr(and)249 2230 y Fk(j)8 b Fn(\021)13 b Fk(\001)e(r)p Fn(\015)g Fk(j)41 b Fr(=)h Fn(\021)570 2237 y Fm(0)601 2230 y Fk(\001)11 b Fn(c)19 b Fr(+)h Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))976 b(\(3.42\))466 2303 y(=)42 b Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(;)203 b Fr(since)31 b Fn(\021)1047 2310 y Fm(0)1078 2303 y Fk(\001)11 b Fn(c)j Fr(=)g(0)p Fn(:)0 2391 y Fr(Th)o(us,)i Fn(\021)d Fk(\001)e(r)p Fn(')16 b Fr(sin)349 2371 y Fm(2)385 2391 y Fn(\022)q(=)p Fr(2)h(is)f(b)q(ounded,)h(and)g(b)o(y)f(virtue)f(of)h(\(3.40\),)125 2463 y Fi(e)122 2480 y Fn(\025)p Fr(\()p Fn(!)r(;)8 b(t;)g(\017)p Fr(\))13 b(=)370 2463 y Fi(e)367 2480 y Fn(\025)395 2487 y Fm(0)415 2480 y Fr(\()p Fn(!)r Fr(\))p Fn(:)e Fr(+)g Fk(O)q Fr(\()p Fk(j)p Fn(t)p Fk(j)p Fr(\))p Fn(:)0 2568 y Fr(This)16 b(implies)e(the)i(lemm)o(a)e(since)i(w)o(e)f(can)i (arbitrarily)e(tak)o(e)1142 2552 y Fi(e)1139 2568 y Fn(\025)1167 2575 y Fm(0)1187 2568 y Fr(\()p Fn(!)r Fr(\))f Fk(\021)g Fr(0.)551 b Fa(2)951 2753 y Fr(29)p eop %%Page: 30 30 30 29 bop 0 -22 a Fc(3.3)70 b(The)22 b(Incoming)g(Outer)g(Solution)0 71 y Fr(W)l(e)15 b(assume)f(that)h(at)g(the)g(initial)e(time)g Fk(\000)p Fn(T)7 b Fr(,)14 b(the)g(w)o(a)o(v)o(e)g(function)h(is)f(giv) o(en)g(b)o(y)g(a)i(semiclassical)c(n)o(uclear)0 131 y(w)o(a)o(v)o(e)23 b(pac)o(k)o(et)g(times)g(an)h(electronic)f(function)h(asso)q(ciated)h (with)f(the)g Fk(B)h Fr(lev)o(el.)43 b(W)l(e)24 b(consider)g(the)0 191 y(asso)q(ciated)17 b(classical)f(quan)o(tities)f(determined)f(b)o (y)h(the)h(follo)o(wing)g(initial)f(conditions)i(at)f Fn(t)e Fr(=)f(0:)133 293 y Fn(a)159 272 y Ff(B)185 293 y Fr(\(0\))42 b(=)g(0)p Fn(;)133 365 y(\021)159 345 y Ff(B)185 365 y Fr(\(0\))g(=)g Fn(\021)395 345 y Fm(0)414 365 y Fn(;)126 438 y(S)159 418 y Ff(B)185 438 y Fr(\(0\))g(=)g(0)p Fn(;)122 511 y(A)159 490 y Ff(B)185 511 y Fr(\(0\))g(=)g Fn(A)406 518 y Fm(0)425 511 y Fn(;)300 b Fr(and)126 615 y Fn(B)166 594 y Ff(B)191 615 y Fr(\()p Fn(t)p Fr(\))41 b Fk(\030)h Fn(B)406 622 y Fm(0)445 615 y Fr(+)507 581 y Fn(i)p Fk(j)p Fn(d)563 588 y Fm(3)583 581 y Fk(j)p 507 603 90 2 v 509 649 a Fn(b)530 656 y Fm(1)549 649 y Fn(\021)575 632 y Fm(0)573 660 y(1)610 542 y Fi(")634 615 y Fn(M)5 b(A)723 622 y Fm(0)762 615 y Fr(+)19 b(ln)877 542 y Fi( )914 581 y Fr(2)p Fn(b)959 588 y Fm(1)979 581 y Fn(\021)1005 563 y Fm(0)1003 593 y(1)1025 581 y Fk(j)p Fn(t)p Fk(j)p 914 603 156 2 v 946 649 a(j)p Fn(d)985 656 y Fm(3)1005 649 y Fk(j)p Fn(\017)1075 542 y Fi(!)1116 615 y Fn(N)5 b(A)1197 622 y Fm(0)1217 542 y Fi(#)1249 615 y Fn(;)0 745 y Fr(where)14 b Fn(A)176 752 y Fm(0)210 745 y Fr(and)g Fn(B)339 752 y Fm(0)374 745 y Fr(satisfy)g(the)g(h)o(yp) q(otheses)h(in)f(the)g(de\014nition)f(of)i(the)f(functions)g Fn(')1580 752 y Fj(l)1593 745 y Fr(,)h(and)g(the)f(asymp-)0 805 y(totic)j(condition)f(for)i Fn(B)445 787 y Ff(B)470 805 y Fr(\()p Fn(t)p Fr(\))f(is)g(to)g(hold)g(for)h(small)d(v)m(alues)i (of)g Fn(\017)p Fr(,)g Fk(j)p Fn(t)p Fk(j)p Fr(,)f Fn(\017=)p Fk(j)p Fn(t)p Fk(j)p Fr(,)g(and)h Fk(j)p Fn(t)g Fr(ln)1599 783 y Fm(2)1636 805 y Fn(\017)8 b Fk(j)p Fr(.)23 b(Aw)o(a)o(y)16 b(from)0 865 y(the)g(a)o(v)o(oided)f(crossing)h(of)g(the)g(electronic)e (lev)o(els,)g(the)h(solution)i(of)f(the)f(Sc)o(hr\177)-24 b(odinger)16 b(equation)g(is)f(w)o(ell)0 925 y(appro)o(ximated)h(b)o(y) h(standard)i(time{dep)q(enden)o(t)d(Born{Opp)q(enheimer)g(w)o(a)o(v)o (e)g(pac)o(k)o(ets.)25 b(Close)17 b(to)h(the)0 985 y(a)o(v)o(oided)i (crossing)h(these)g(standard)h(w)o(a)o(v)o(e)d(pac)o(k)o(ets)h(fail)g (to)h(appro)o(ximate)e(the)i(solution.)35 b(The)21 b(next)0 1046 y(lemma)15 b(tells)j(us)g(ho)o(w)h(close)f(to)h(the)f(a)o(v)o (oided)g(crossing)h(time)d(these)i(standard)h(w)o(a)o(v)o(e)f(pac)o(k)o (ets)f(can)i(b)q(e)0 1106 y(used)d(as)h(appro)o(ximations.)0 1220 y Fg(Lemma)f(3.3)24 b Fl(Supp)n(ose)18 b Fr(0)d Fn(<)g(\016)597 1202 y Ff(0)623 1220 y Fn(<)g(\030)j(<)d Fr(1)p Fl(.)24 b(F)l(or)18 b(any)g Fk(\000)p Fn(T)j Fk(\024)15 b Fn(t)f Fk(\024)h(\000)p Fn(\017)1305 1202 y Fm(1)p Ff(\000)p Fj(\030)1368 1220 y Fl(,)j(ther)n(e)g(is)g(an)h(appr)n (oximation)0 1280 y Fn( )32 1287 y Fj(O)q(I)79 1280 y Fr(\()p Fn(x;)8 b(t)p Fr(\))17 b Fl(to)h(the)f(exact)i(solution)g Fn( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))16 b Fl(to)h(the)h(Schr\177)-25 b(odinger)18 b(e)n(quation,)h(such)e(that)122 1413 y Fn( )154 1420 y Fj(O)q(I)201 1413 y Fr(\()p Fn(x;)8 b(t)p Fr(\))13 b(=)h Fn(F)419 1340 y Fi( )457 1379 y Fk(k)p Fn(x)c Fk(\000)h Fn(a)596 1361 y Ff(B)622 1379 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p 457 1401 246 2 v 532 1447 a Fn(\017)552 1433 y Fm(1)p Ff(\000)p Fj(\016)614 1423 y Fe(0)707 1340 y Fi(!)748 1413 y Fn(e)771 1392 y Fj(iS)806 1381 y Fe(B)829 1392 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)902 1381 y Fd(2)921 1413 y Fn(')953 1420 y Fj(l)966 1413 y Fr(\()p Fn(A)1022 1392 y Ff(B)1047 1413 y Fr(\()p Fn(t)p Fr(\))p Fn(;)d(B)1165 1392 y Ff(B)1190 1413 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1288 1392 y Fm(2)1307 1413 y Fn(;)g(a)1355 1392 y Ff(B)1380 1413 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1484 1392 y Ff(B)1510 1413 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))16 b(\010)1686 1392 y Ff(\000)1686 1425 y(B)1716 1413 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))1825 1517 y(\(3.43\))0 1618 y Fl(and)122 1720 y Fn( )r Fr(\()p Fn(x;)g(t)p Fr(\))k(=)i Fn( )358 1727 y Fj(O)q(I)406 1720 y Fr(\()p Fn(x;)8 b(t)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\017)651 1700 y Fj(\030)669 1720 y Fr(\))p Fn(;)1123 b Fr(\(3.44\))0 1822 y Fl(in)18 b(the)g Fn(L)174 1804 y Fm(2)194 1822 y Fr(\()p Fn(I)-10 b(R)262 1801 y Fj(n)285 1822 y Fr(\))18 b Fl(sense,)g(as)f Fn(\017)d Fk(!)g Fr(0)p Fl(.)0 1936 y Fg(Pro)r(of:)0 1996 y Fr(The)j(pro)q(of)g(of)g(this)f(Lemma)f(is)h(v)o(ery)f(similar)g (to)i(that)g(of)f(Lemma)f(6.4)i(of)g([17].)k(F)l(or)c Fk(\000)p Fn(T)k Fk(\024)14 b Fn(t)g Fk(\024)g(\000)p Fn(\017)1887 1978 y Fm(1)p Ff(\000)p Fj(\030)0 2056 y Fr(and)h Fn(x)g Fr(in)f(the)g(supp)q(ort)i(of)f(the)f(cut)h(o\013)g (function)g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)h Fk(\000)g Fn(a)1120 2038 y Ff(B)1144 2056 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)1269 2038 y Fm(1)p Ff(\000)p Fj(\016)1331 2027 y Fe(0)1343 2056 y Fr(\),)1513 2037 y Fm(1)p 1411 2045 224 2 v 1411 2074 a Fj(E)1437 2080 y Fe(A)1463 2074 y Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(E)1564 2080 y Fe(B)1587 2074 y Fm(\()p Fj(x)p Fm(\))1654 2056 y Fr(is)14 b(b)q(ounded)i(b)o(y)0 2127 y(a)i(m)o(ultiple)c(of)j Fn(\017)310 2109 y Ff(\000)p Fm(1+)p Fj(\030)401 2127 y Fr(.)25 b(F)l(urthermore,)15 b(for)j(suc)o(h)f Fn(x)g Fr(and)h Fn(t)p Fr(,)e(a)i(rather)f(tedious)h (calculation)e(sho)o(ws)j(that)0 2139 y Fi(\020)30 2168 y Fj(@)p 30 2176 34 2 v 30 2205 a(@)r(t)79 2187 y Fr(+)11 b Fn(\021)154 2169 y Ff(B)191 2187 y Fk(\001)g(r)258 2194 y Fj(x)279 2139 y Fi(\021)321 2187 y Fr(\010)356 2167 y Ff(\000)356 2200 y(B)386 2187 y Fr(\()p Fn(x;)d(t;)g(\017)p Fr(\))13 b(is)i(b)q(ounded)h(b)o(y)e(a)i(m)o(ultiple)c(of)j Fn(\017)1163 2169 y Ff(\000)p Fm(1+)p Fj(\030)1254 2187 y Fr(.)21 b(Th)o(us,)15 b(up)g(to)g(errors)h(on)f(the)g(order)0 2258 y(of)i Fn(\017)76 2239 y Fm(2)p Fj(\030)112 2258 y Fr(,)32 b Fn( )190 2265 y Fj(O)q(I)237 2258 y Fr(\()p Fn(x;)8 b(t)p Fr(\))16 b(is)g(equal)f(to)122 2365 y(\011\()p Fn(x;)8 b(t)p Fr(\))13 b(=)h Fn(F)7 b Fr(\()p Fk(k)p Fn(x)j Fk(\000)g Fn(a)527 2345 y Ff(B)553 2365 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)678 2345 y Fm(1)p Ff(\000)p Fj(\016)740 2333 y Fe(0)753 2365 y Fr(\))e Fn(e)803 2345 y Fj(iS)838 2333 y Fe(B)861 2345 y Fm(\()p 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335 y Fr(W)l(e)i(mak)o(e)d(a)k(sequence)d(of)i (replacemen)o(ts)e(in)h(form)o(ula)f(\(3.43\).)22 b(With)14 b(eac)o(h)h(replacemen)o(t,)c(w)o(e)k(mak)o(e)e(an)0 395 y(error)j(that)h(is)f(acceptably)f(small)g(b)q(ecause)i(of)f (earlier)f(results.)73 455 y(W)l(e)h(in)o(tro)q(duce)g(the)g(notation)h Fn(y)679 437 y Ff(C)715 455 y Fr(=)d(\()p Fn(x)c Fk(\000)h Fn(a)900 437 y Ff(C)922 455 y Fr(\()p Fn(t)p Fr(\)\))p Fn(=\017)p Fr(,)16 b(and)h(using)f(prop)q(osition)i(2.2,)e(w)o(e)g (replace)122 547 y Fn(F)7 b Fr(\()p Fn(\017)200 526 y Fj(\016)217 515 y Fe(0)229 547 y Fk(k)p Fn(y)280 526 y Ff(B)306 547 y Fk(k)p Fr(\))97 b(b)o(y)g Fn(F)7 b Fr(\()p Fn(\017)674 526 y Fj(\016)691 515 y Fe(0)703 547 y Fk(k)p Fn(y)r Fk(k)p Fr(\))p Fn(:)0 639 y Fr(Next)15 b(w)o(e)h(use)g(prop)q (osition)i(2.2)e(and)h(corollary)f(2.2)h(to)f(replace)f(the)h(factor) 205 759 y(exp)288 686 y Fi( )321 759 y Fn(i)343 725 y(\021)369 707 y Ff(B)394 725 y Fr(\()p Fn(t)p Fr(\))11 b Fk(\001)g Fn(y)512 707 y Ff(B)p 343 747 196 2 v 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Fn(b)1623 873 y Fm(1)1643 866 y Fn(\021)1669 848 y Fm(0)1667 879 y(1)1689 866 y Fk(j)p Fn(t)p Fk(j)p 1578 888 156 2 v 1610 934 a(j)p Fn(d)1649 941 y Fm(3)1669 934 y Fk(j)p Fn(\017)1750 900 y Fr(+)1804 866 y(1)p 1804 888 25 2 v 1804 934 a(2)1833 827 y Fi(!!)1907 900 y Fn(:)0 1023 y Fr(W)l(e)16 b(use)g(lemma)d(3.1)k(of)g([20])f(and)h(prop)q(osition)g (2.2)f(to)h(replace)122 1115 y Fn(')154 1122 y Fj(l)167 1115 y Fr(\()p Fn(A)223 1094 y Ff(B)248 1115 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)366 1094 y Ff(B)391 1115 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)489 1094 y Fm(2)508 1115 y Fn(;)g(a)556 1094 y Ff(B)582 1115 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(0)p Fn(;)g(x)p Fr(\))97 b(b)o(y)g Fn(')1031 1122 y Fj(l)1044 1115 y Fr(\()p Fn(A)1100 1094 y Ff(B)1125 1115 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1243 1094 y Ff(B)1268 1115 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1366 1094 y Fm(2)1385 1115 y Fn(;)g(a)p Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(0)p Fn(;)g(x)p Fr(\))p Fn(:)0 1206 y Fr(W)l(e)16 b(then)g(use)g(lemma)e(3.1)i(of)h([20])f(and)h(prop)q(osition)g(2.3)g (to)f(replace)205 1329 y Fn(')237 1336 y Fj(l)250 1329 y Fr(\()p Fn(A)306 1309 y Ff(B)331 1329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)449 1309 y Ff(B)474 1329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)572 1309 y Fm(2)591 1329 y Fn(;)g(a)p Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(0)p Fn(;)g(x)p Fr(\))97 b(b)o(y)205 1470 y(exp)288 1397 y Fi( )321 1470 y Fk(\000)p Fn(i)400 1437 y Fk(j)p Fn(d)439 1444 y Fm(3)459 1437 y Fk(j)p 382 1459 111 2 v 382 1504 a Fr(2)p Fn(b)427 1511 y Fm(1)446 1504 y Fn(\021)472 1487 y Fm(0)470 1515 y(1)513 1470 y Fk(h)8 b Fn(y)r(;)596 1397 y Fi(")629 1470 y Fn(M)24 b Fr(+)c(ln)815 1397 y Fi( )853 1437 y Fr(2)p Fn(b)898 1444 y Fm(1)918 1437 y Fn(\021)944 1419 y Fm(0)942 1449 y(1)963 1437 y Fk(j)p Fn(t)p Fk(j)p 853 1459 156 2 v 884 1504 a(j)p Fn(d)923 1511 y Fm(3)943 1504 y Fk(j)p Fn(\017)1013 1397 y Fi(!)1063 1470 y Fn(N)1115 1397 y Fi(#)1156 1470 y Fn(y)r Fk(i)1217 1397 y Fi(!)1275 1470 y Fn(\017)1295 1450 y Ff(\000)p Fj(n=)p Fm(2)1389 1470 y Fn(')1421 1477 y Fj(l)1434 1470 y Fr(\()p Fn(A)1490 1477 y Fm(0)1509 1470 y Fn(;)8 b(B)1568 1477 y Fm(0)1588 1470 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))p Fn(:)0 1590 y Fr(By)16 b(corollary)f(2.3)i(w)o(e)f(can)g (replace)122 1710 y(exp)205 1637 y Fi( )238 1710 y Fn(i)260 1677 y(S)293 1659 y Ff(B)318 1677 y Fr(\()p Fn(t)p Fr(\))p 260 1699 115 2 v 297 1745 a Fn(\017)317 1730 y Fm(2)379 1637 y Fi(!)517 1710 y Fr(b)o(y)106 b(exp)758 1637 y Fi( )790 1710 y Fn(i)812 1677 y(S)s Fr(\()p Fn(t)p Fr(\))p 812 1699 89 2 v 837 1745 a Fn(\017)857 1730 y Fm(2)917 1710 y Fr(+)11 b Fn(i)988 1677 y(S)1021 1659 y Ff(B)1018 1689 y Fm(0)1046 1677 y Fr(\()p Fn(\017;)d Fk(\000)p Fr(\))p 988 1699 178 2 v 1056 1745 a Fn(\017)1076 1730 y Fm(2)1180 1710 y Fk(\000)j Fn(i)1252 1677 y Fk(j)p Fn(d)1291 1684 y Fm(3)1311 1677 y Fk(j)p Fn(t)p 1252 1699 91 2 v 1287 1745 a(\017)1358 1710 y Fk(\000)g Fn(i)1430 1677 y(b)1451 1684 y Fm(1)1470 1677 y Fn(\021)1496 1659 y Fm(0)1494 1689 y(1)1516 1677 y Fn(t)1534 1659 y Fm(2)p 1430 1699 124 2 v 1472 1745 a Fn(\017)1492 1730 y Fm(2)1569 1710 y Fk(\000)g Fn(i)1641 1677 y(d)1666 1659 y Fm(2)1666 1689 y(3)1694 1677 y Fr(ln)16 b Fk(j)p Fn(t)p Fk(j)p 1641 1699 156 2 v 1675 1745 a Fn(\021)1701 1727 y Fm(0)1699 1756 y(1)1721 1745 y Fn(b)1742 1752 y Fm(1)1801 1637 y Fi(!)1843 1710 y Fn(:)0 1831 y Fr(Finally)l(,)e(using)j(lemmas)c(3.1) k(and)g(3.2,)f(w)o(e)g(replace)122 1922 y(\010)157 1902 y Ff(\000)157 1935 y(B)187 1922 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))96 b(b)o(y)108 b Fk(\000)19 b Fn( )681 1929 y Fm(1)701 1922 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))0 2014 y(The)16 b(lemma)d(follo)o(ws.)1498 b Fa(2)0 2217 y Fc(3.4)70 b(The)22 b(Inner)h(Solution)0 2309 y Fr(F)l(or)13 b(small)e(times)f(w)o (e)i(use)h(a)g(di\013eren)o(t)f(appro)o(ximation)f(that)i(is)f (constructed)h(b)o(y)f(means)g(of)g(the)h(classical)0 2369 y(quan)o(tities)20 b(asso)q(ciated)j(with)e(the)g(p)q(oten)o(tial) 892 2353 y Fi(e)884 2369 y Fn(V)11 b Fr(\()p Fn(x;)d(\017)p Fr(\).)36 b(Let)21 b Fn(a)p Fr(\()p Fn(t)p Fr(\),)h Fn(\021)r Fr(\()p Fn(t)p Fr(\),)f(and)h Fn(S)s Fr(\()p Fn(t)p Fr(\))f(b)q(e)h (the)f(classical)0 2430 y(quan)o(tities)15 b(that)i(satisfy)f(the)g (initial)f(conditions)141 2512 y Fn(a)p Fr(\(0\))f(=)f(0)130 2573 y Fn(\021)r Fr(\(0\))h(=)g Fn(\021)310 2554 y Fm(0)131 2633 y Fn(S)s Fr(\(0\))g(=)f(0)p Fn(:)1825 2573 y Fr(\(3.51\))951 2753 y(32)p eop %%Page: 33 33 33 32 bop 0 -22 a Fr(In)16 b(the)g(rescaled)f(v)m(ariables)122 31 y Fi(\032)161 61 y Fn(y)h Fr(=)e(\()p Fn(x)c Fk(\000)h Fn(a)p Fr(\()p Fn(t)p Fr(\)\))p Fn(=\017;)258 122 y(s)j Fr(=)f Fn(t=\017;)1825 92 y Fr(\(3.52\))0 214 y(the)j(Sc)o(hr\177)-24 b(odinger)16 b(equation)g(is)122 343 y Fn(i\017)175 309 y(@)p 164 331 52 2 v 164 377 a(@)s(s)220 343 y( )c Fk(\000)f Fn(i\017\021)r Fr(\()p Fn(\017s)p Fr(\))p Fk(r)500 350 y Fj(y)519 343 y Fn( )k Fr(=)f Fk(\000)662 309 y Fn(\017)682 291 y Fm(2)p 662 331 40 2 v 669 377 a Fr(2)706 343 y Fk(4)751 350 y Fj(y)771 343 y Fn( )f Fr(+)e Fn(h)p Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))f(+)h Fn(\017y)r(;)d(\017)p Fr(\))p Fn( )r(:)592 b Fr(\(3.53\))0 457 y(W)l(e)16 b(lo)q(ok)h(for)f (an)h(appro)o(ximate)e(solution)h(of)h(the)f(form)122 587 y Fn( )r Fr(\()p Fn(y)r(;)8 b(s;)g(\017)p Fr(\))28 b(=)i Fn(F)7 b Fr(\()p Fk(k)p Fn(y)r Fk(k)p Fn(\017)557 567 y Fj(\016)574 555 y Fe(0)586 587 y Fr(\))h(exp)696 514 y Fi( )729 587 y Fn(i)751 554 y(S)s Fr(\()p Fn(\017s)p Fr(\))p 751 576 114 2 v 788 622 a Fn(\017)808 607 y Fm(2)880 587 y Fr(+)j Fn(i)951 554 y(\021)r Fr(\()p Fn(\017s)p Fr(\))p Fn(y)p 951 576 133 2 v 1007 622 a(\017)1088 514 y Fi(!)1129 587 y Fn(\037)p Fr(\()p Fn(y)r(;)d(s;)g(\017)p Fr(\))p Fn(;)500 b Fr(\(3.54\))0 717 y(with)122 819 y Fn(\037)p Fr(\()p Fn(y)r(;)8 b(s;)g(\017)p Fr(\))k(=)83 b Fk(f)p Fn(f)5 b Fr(\()p Fn(y)r(;)j(s;)g(\017)p Fr(\))p Fn( )674 826 y Fm(1)693 819 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017)p Fr(\))i(+)h Fn(g)r Fr(\()p Fn(y)r(;)d(s;)g(\017)p Fr(\))p Fn( )1252 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2085 y Fi(X)451 2176 y Fj(j)r Fm(=0)520 2127 y Fn(\027)544 2134 y Fj(j)563 2127 y Fr(\()p Fn(\017)p Fr(\))p Fn( )653 2134 y Ff(?)p Fj(j)698 2127 y Fr(\()p Fn(x;)8 b(t)p Fr(\))p Fn(;)1007 b Fr(\(3.57\))0 2270 y(with)18 b(asymptotic)e(scales)i Fn(\027)527 2277 y Fj(j)545 2270 y Fr(\()p Fn(\017)p Fr(\))g(to)g(b)q(e)g(determined)d(b)o(y)j(matc)o(hing.)24 b(W)l(e)18 b(insert)f(these)h(expansions)g(in)0 2330 y(\(3.56\).)k(Using)16 b(the)g(b)q(eha)o(viors)g(\(1.33\))h(and)g (lemma)c(2.2)k(w)o(e)f(see)g(that)g(the)g(lo)o(w)o(est)g(order)g(terms) f(yield)243 2456 y Fn(i\017\027)304 2463 y Fm(0)323 2456 y Fr(\()p Fn(\017)p Fr(\))397 2423 y Fn(@)p 386 2445 V 386 2491 a(@)s(s)442 2456 y(f)466 2463 y Fm(0)486 2456 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))p Fn( )627 2463 y Fm(1)646 2456 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017)p Fr(\))i(+)h Fn(i\017\027)1058 2463 y Fm(0)1077 2456 y Fr(\()p Fn(\017)p Fr(\))1151 2423 y Fn(@)p 1140 2445 V 1140 2491 a(@)s(s)1196 2456 y(g)1219 2463 y Fm(0)1239 2456 y Fr(\()p Fn(y)r(;)d(s)p Fr(\))p Fn( )1380 2463 y Fm(2)1399 2456 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017)p Fr(\))122 2545 y(=)83 b Fn(h)271 2552 y Ff(?)300 2545 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))11 b(+)g Fn(\017y)r(;)d(\017)p Fr(\))p Fn( )625 2552 y Ff(?)653 2545 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017s;)g(\017)p Fr(\))243 2617 y(+)p Fn(\017\027)325 2624 y Fm(0)344 2617 y Fr(\()p Fn(\017)p Fr(\))p Fn(h)430 2624 y Fm(11)467 2617 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))g(\()p Fn(f)627 2624 y Fm(0)647 2617 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))p Fn( )788 2624 y Fm(1)807 2617 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017)p Fr(\))i(+)h Fn(g)1181 2624 y Fm(0)1201 2617 y Fr(\()p Fn(y)r(;)d(s)p Fr(\))p Fn( )1342 2624 y Fm(2)1360 2617 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))j(+)g Fn(\017y)r(;)d(\017)p Fr(\)\))153 b(\(3.58\))951 2753 y(33)p eop %%Page: 34 34 34 33 bop 0 -22 a Fr(on)25 b(the)f(supp)q(ort)h(of)f Fn(F)7 b Fr(,)25 b(where)f Fn(h)674 -15 y Fm(11)712 -22 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))23 b(is)h(the)g(op)q(erator)h(on)g (the)f(span)h(of)f Fk(f)8 b Fn( )1615 -15 y Fm(1)1635 -22 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))15 b(+)i Fn(\017y)r(;)8 b(\017)p Fr(\),)0 39 y Fn( )32 46 y Fm(2)51 39 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))g(+)g Fn(\017y)r(;)g(\017) p Fr(\))g Fk(g)14 b Fr(whose)h(matrix)e(in)i(the)f(basis)h Fk(f)8 b Fn( )1005 46 y Fm(1)1025 39 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))g(+)g Fn(\017y)r(;)g(\017)p Fr(\))p Fn(;)14 b( )1372 46 y Fm(2)1392 39 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))8 b(+)g Fn(\017y)r(;)g(\017)p Fr(\))g Fk(g)13 b Fr(is)i(giv)o(en)f(b)o(y)122 93 y Fi(\022)161 124 y Fn(b)182 131 y Fm(1)201 124 y Fn(\021)227 105 y Fm(0)225 136 y(1)247 124 y Fn(s)d Fr(+)g Fn(b)351 131 y Fm(1)370 124 y Fn(y)394 131 y Fm(1)425 124 y Fr(+)g Fn(b)495 131 y Fm(2)514 124 y Fn(y)538 131 y Fm(2)722 124 y Fn(c)743 131 y Fm(2)763 124 y Fn(y)787 131 y Fm(2)818 124 y Fr(+)g 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Fk(\001)g(\001)g(\001)g Fn(;)g(m)i Fk(\000)g Fr(1)p Fn(:)15 b Fr(The)h(sp)q(ectrum)f(of)h Fn(h)1234 455 y Ff(?)1263 448 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))15 b(is)h(b)q(ounded)g(a)o(w)o(a)o(y)g(from)e(0)i(in)0 508 y(a)h(neigh)o(b)q(orho)q(o)q(d)h(of)e(\(0)p Fn(;)8 b Fr(0\).)22 b(This)17 b(implies)122 599 y Fn( )154 606 y Ff(?)p Fj(j)199 599 y Fr(\()p Fn(x;)8 b(t)p Fr(\))13 b(=)h(0)p Fn(;)50 b(j)17 b Fr(=)c(0)p Fn(;)8 b Fr(1)p Fn(;)g Fk(\001)g(\001)g(\001)h Fn(;)f(m)j Fk(\000)g Fr(1)p Fn(:)956 b Fr(\(3.60\))0 691 y(By)24 b(pro)s(jecting)f(with)i Fn(P)7 b Fr(\()p Fn(x;)h(\017)p Fr(\))23 b(and)i(\()p Fn(I)-34 b(I)20 b Fk(\000)c Fn(P)7 b Fr(\()p Fn(x;)h(\017)p Fr(\)\),)26 b(w)o(e)e(split)f(the)h(remaining)f(equation)h(for)h(order) 0 751 y Fn(\017\027)44 758 y Fm(0)64 751 y Fr(\()p Fn(\017)p Fr(\))13 b(=)h Fn(\027)211 758 y Fj(m)244 751 y Fr(\()p Fn(\017)p Fr(\),)243 867 y Fn(i\017\027)304 874 y Fm(0)323 867 y Fr(\()p Fn(\017)p Fr(\))397 833 y Fn(@)p 386 855 52 2 v 386 901 a(@)s(s)442 867 y(f)466 874 y Fm(0)486 867 y Fn( )518 874 y Fm(1)548 867 y Fr(+)d Fn(i\017\027)658 874 y Fm(0)677 867 y Fr(\()p Fn(\017)p Fr(\))752 833 y Fn(@)p 740 855 V 740 901 a(@)s(s)797 867 y(g)820 874 y Fm(0)840 867 y Fn( )872 874 y Fm(2)122 955 y Fr(=)83 b Fn(\027)267 962 y Fj(m)300 955 y Fr(\()p Fn(\017)p Fr(\))p Fn(h)386 962 y Ff(?)416 955 y Fn( )448 962 y Ff(?)488 955 y Fr(+)11 b Fn(\017\027)581 962 y Fm(0)600 955 y Fr(\()p Fn(\017)p Fr(\))p Fn(h)686 962 y Fm(11)723 955 y Fr(\()p Fn(f)766 962 y Fm(0)786 955 y Fn( )818 962 y Fm(1)849 955 y Fr(+)g Fn(g)921 962 y Fm(0)941 955 y Fn( )973 962 y Fm(2)992 955 y Fr(\))p Fn(;)800 b Fr(\(3.61\))0 1046 y(in)o(to)122 1149 y Fn(i\017\027)183 1156 y Fm(0)202 1149 y Fr(\()p Fn(\017)p Fr(\))276 1116 y Fn(@)p 265 1138 V 265 1183 a(@)s(s)321 1149 y(f)345 1156 y Fm(0)365 1149 y Fn( )397 1156 y Fm(1)427 1149 y Fr(+)11 b Fn(i\017\027)537 1156 y Fm(0)557 1149 y Fr(\()p Fn(\017)p Fr(\))631 1116 y Fn(@)p 620 1138 V 620 1183 a(@)s(s)676 1149 y(g)699 1156 y Fm(0)719 1149 y Fn( )751 1156 y Fm(2)812 1149 y Fr(=)41 b Fn(\017\027)935 1156 y Fm(0)955 1149 y Fr(\()p Fn(\017)p Fr(\))p Fn(h)1041 1156 y Fm(11)1078 1149 y Fr(\()p Fn(f)1121 1156 y Fm(0)1140 1149 y Fn( )1172 1156 y Fm(1)1203 1149 y Fr(+)11 b Fn(g)1275 1156 y Fm(0)1295 1149 y Fn( )1327 1156 y Fm(2)1346 1149 y Fr(\))98 b(and)505 1250 y Fn(\027)529 1257 y Fj(m)562 1250 y Fr(\()p Fn(\017)p Fr(\))p Fn(h)648 1257 y Ff(?)678 1250 y Fn( )710 1257 y Ff(?)p Fj(m)812 1250 y Fr(=)41 b(0)p Fn(:)896 b Fr(\(3.62\))73 1341 y(The)16 b(second)h(equation)f(giv)o(es)g Fn( )682 1348 y Ff(?)p Fj(m)756 1341 y Fr(=)e(0.)21 b(The)16 b(\014rst)h(one)f (is)g(equiv)m(alen)o(t)f(to)122 1454 y Fn(i)155 1420 y(@)p 144 1442 V 144 1488 a(@)s(s)208 1393 y Fi(\022)247 1424 y Fn(f)271 1431 y Fm(0)291 1424 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))247 1484 y Fn(g)270 1491 y Fm(0)291 1484 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))408 1393 y Fi(\023)452 1454 y Fr(=)504 1393 y Fi(\022)543 1424 y Fn(b)564 1431 y Fm(1)583 1424 y Fn(\021)609 1406 y Fm(0)607 1436 y(1)629 1424 y Fn(s)j Fr(+)g Fn(b)733 1431 y Fm(1)752 1424 y Fn(y)776 1431 y Fm(1)807 1424 y Fr(+)g Fn(b)877 1431 y Fm(2)896 1424 y Fn(y)920 1431 y Fm(2)1104 1424 y Fn(c)1125 1431 y Fm(2)1145 1424 y Fn(y)1169 1431 y Fm(2)1199 1424 y Fr(+)g Fn(id)1290 1431 y Fm(3)638 1484 y Fn(c)659 1491 y Fm(2)678 1484 y Fn(y)702 1491 y Fm(2)733 1484 y Fk(\000)g Fn(id)825 1491 y Fm(3)988 1484 y Fk(\000)p Fn(b)1048 1491 y Fm(1)1068 1484 y Fn(\021)1094 1466 y Fm(0)1092 1496 y(1)1113 1484 y Fn(s)g Fk(\000)g Fn(b)1218 1491 y Fm(1)1237 1484 y Fn(y)1261 1491 y Fm(1)1292 1484 y Fk(\000)g Fn(b)1363 1491 y Fm(2)1382 1484 y Fn(y)1406 1491 y Fm(2)1434 1393 y Fi(\023)d(\022)1512 1424 y Fn(f)1536 1431 y Fm(0)1556 1424 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))1512 1484 y Fn(g)1535 1491 y Fm(0)1555 1484 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))1672 1393 y Fi(\023)1711 1454 y Fn(:)100 b Fr(\(3.63\))0 1566 y(The)20 b(general)g(solution)h(of)f(this)h(equation)f(can)g(b)q (e)h(found)g(exactly)e(in)h(terms)e(of)j(parab)q(olic)g(cylinder)0 1626 y(functions)16 b([6].)130 1726 y Fi(\022)169 1756 y Fn(f)193 1763 y Fm(0)213 1756 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))169 1817 y Fn(g)192 1824 y Fm(0)213 1817 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))329 1726 y Fi(\023)402 1786 y Fr(=)95 b Fn(C)570 1793 y Fm(1)590 1786 y Fr(\()p Fn(y)r Fr(\))662 1676 y Fi(0)662 1749 y(B)662 1774 y(B)662 1801 y(@)711 1707 y Fm(\(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)825 1712 y Fd(2)842 1707 y Fj(y)859 1712 y Fd(2)877 1707 y Fm(+)p Fj(id)934 1712 y Fd(3)952 1707 y Fm(\))p 711 1718 255 2 v 774 1757 a(2)792 1724 y Fk(p)p 834 1724 69 2 v 33 x Fj(b)849 1762 y Fd(1)866 1757 y Fj(\021)885 1746 y Fd(0)884 1768 y(1)970 1730 y Fn(D)1010 1737 y Fj(p)1028 1741 y Fh(u)1049 1737 y Ff(\000)p Fm(1)1105 1669 y Fi(\022)1140 1707 y Fm(\()p Ff(\000)p Fm(1+)p Fj(i)p Fm(\))p 1140 1718 112 2 v 1141 1724 a Fk(p)p 1183 1724 69 2 v 33 x Fj(b)1198 1762 y Fd(1)1215 1757 y Fj(\021)1234 1746 y Fd(0)1233 1768 y(1)1257 1730 y Fr(\()p Fn(b)1297 1737 y Fm(1)1317 1730 y Fn(\021)1343 1712 y Fm(0)1341 1742 y(1)1362 1730 y Fn(s)11 b Fr(+)g Fn(b)1466 1737 y Fm(1)1486 1730 y Fn(y)1510 1737 y Fm(1)1540 1730 y Fr(+)g Fn(b)1610 1737 y Fm(2)1630 1730 y Fn(y)1654 1737 y Fm(2)1673 1730 y Fr(\))1692 1669 y Fi(\023)861 1838 y Fn(D)901 1845 y Fj(p)919 1849 y Fh(u)950 1778 y Fi(\022)985 1815 y Fm(\()p Ff(\000)p Fm(1+)p Fj(i)p Fm(\))p 985 1827 112 2 v 986 1832 a Fk(p)p 1028 1832 69 2 v 33 x Fj(b)1043 1870 y Fd(1)1060 1865 y Fj(\021)1079 1854 y Fd(0)1078 1877 y(1)1102 1838 y Fr(\()p Fn(b)1142 1845 y Fm(1)1162 1838 y Fn(\021)1188 1820 y Fm(0)1186 1851 y(1)1207 1838 y Fn(s)g Fr(+)h Fn(b)1312 1845 y Fm(1)1331 1838 y Fn(y)1355 1845 y Fm(1)1386 1838 y Fr(+)f Fn(b)1456 1845 y Fm(2)1475 1838 y Fn(y)1499 1845 y Fm(2)1518 1838 y Fr(\))1537 1778 y Fi(\023)1731 1676 y(1)1731 1749 y(C)1731 1774 y(C)1731 1801 y(A)1825 1786 y Fr(\(3.64\))481 2025 y(+)16 b Fn(C)570 2032 y Fm(2)590 2025 y Fr(\()p Fn(y)r Fr(\))678 1915 y Fi(0)678 1988 y(B)678 2013 y(B)678 2039 y(@)891 1969 y Fn(D)931 1976 y Fj(p)949 1982 y Fh(l)972 1908 y Fi(\022)1007 1945 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1007 1957 112 2 v 1008 1963 a Fk(p)p 1050 1963 69 2 v 33 x Fj(b)1065 2001 y Fd(1)1082 1996 y Fj(\021)1101 1984 y Fd(0)1100 2007 y(1)1124 1969 y Fr(\()p Fn(b)1164 1976 y Fm(1)1184 1969 y Fn(\021)1210 1951 y Fm(0)1208 1981 y(1)1229 1969 y Fn(s)11 b Fr(+)g Fn(b)1333 1976 y Fm(1)1353 1969 y Fn(y)1377 1976 y Fm(1)1407 1969 y Fr(+)g Fn(b)1477 1976 y Fm(2)1497 1969 y Fn(y)1521 1976 y Fm(2)1540 1969 y Fr(\))1559 1908 y Fi(\023)727 2054 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)868 2059 y Fd(2)886 2054 y Fj(y)903 2059 y Fd(2)921 2054 y Ff(\000)p Fj(id)978 2059 y Fd(3)995 2054 y Fm(\))p 727 2065 282 2 v 804 2104 a(2)822 2071 y Fk(p)p 864 2071 69 2 v 33 x Fj(b)879 2109 y Fd(1)896 2104 y Fj(\021)915 2093 y Fd(0)914 2115 y(1)1014 2077 y Fn(D)1054 2084 y Fj(p)1072 2090 y Fh(l)1084 2084 y Ff(\000)p Fm(1)1140 2016 y Fi(\022)1175 2054 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1175 2065 112 2 v 1176 2071 a Fk(p)p 1218 2071 69 2 v 33 x Fj(b)1233 2109 y Fd(1)1250 2104 y Fj(\021)1269 2093 y Fd(0)1268 2115 y(1)1292 2077 y Fr(\()p Fn(b)1332 2084 y Fm(1)1352 2077 y Fn(\021)1378 2059 y Fm(0)1376 2089 y(1)1397 2077 y Fn(s)g Fr(+)g Fn(b)1501 2084 y Fm(1)1521 2077 y Fn(y)1545 2084 y Fm(1)1575 2077 y Fr(+)h Fn(b)1646 2084 y Fm(2)1665 2077 y Fn(y)1689 2084 y Fm(2)1708 2077 y Fr(\))1727 2016 y Fi(\023)1766 1915 y(1)1766 1988 y(C)1766 2013 y(C)1766 2039 y(A)1823 2025 y Fn(;)0 2182 y Fr(where)122 2291 y Fn(p)146 2298 y Fj(l)190 2291 y Fr(=)30 b Fk(\000)p Fn(p)321 2298 y Fj(u)373 2291 y Fr(=)g Fk(\000)16 b Fn(i)526 2257 y(c)547 2239 y Fm(2)547 2269 y(2)567 2257 y Fn(y)593 2239 y Fm(2)591 2269 y(2)623 2257 y Fr(+)11 b Fn(d)697 2239 y Fm(2)697 2269 y(3)p 526 2279 192 2 v 567 2325 a Fr(2)p Fn(b)612 2332 y Fm(1)632 2325 y Fn(\021)658 2308 y Fm(0)656 2336 y(1)722 2291 y Fn(:)0 2405 y Fr(W)l(e)16 b(de\014ne)g(our)h(inner)e(appro)o(ximation)g(b)o(y)122 2525 y Fn( )154 2532 y Fj(I)174 2525 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))13 b(=)41 b Fn(F)48 b Fr(\()p Fk(k)p Fn(y)r Fk(k)p Fn(\017)570 2504 y Fj(\016)587 2493 y Fe(0)600 2525 y Fr(\))24 b(exp)726 2452 y Fi( )759 2525 y Fn(i)781 2491 y(S)s Fr(\()p Fn(\017s)p Fr(\))p 781 2513 114 2 v 818 2559 a Fn(\017)838 2545 y Fm(2)910 2525 y Fr(+)11 b Fn(i)981 2491 y(\021)r Fr(\()p Fn(\017s)p Fr(\))p Fn(y)p 981 2513 133 2 v 1037 2559 a(\017)1118 2452 y Fi(!)1825 2525 y Fr(\(3.65\))375 2629 y Fk(\002)41 b Fr([)8 b Fn(\027)501 2636 y Fm(0)521 2629 y Fr(\()p Fn(\017)p Fr(\))p Fn(f)603 2636 y Fm(0)623 2629 y Fr(\()p Fn(y)r(;)g(s)p Fr(\))p Fn( )764 2636 y Fm(1)782 2629 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))j(+)g Fn(\017y)r(;)d(\017)p Fr(\))h(+)i Fn(\027)1157 2636 y Fm(0)1177 2629 y Fr(\()p Fn(\017)p Fr(\))p Fn(g)1258 2636 y Fm(0)1278 2629 y Fr(\()p Fn(y)r(;)d(s)p Fr(\))p Fn( )1419 2636 y Fm(2)1438 2629 y Fr(\()p Fn(a)p Fr(\()p Fn(\017s)p Fr(\))i(+)h Fn(\017y)r(;)d(\017)p Fr(\))g(])f Fn(;)951 2753 y Fr(34)p eop %%Page: 35 35 35 34 bop 0 -22 a Fr(with)16 b Fn(f)135 -15 y Fm(0)171 -22 y Fr(and)h Fn(g)289 -15 y Fm(0)325 -22 y Fr(as)g(ab)q(o)o(v)o(e.)k (T)l(o)c(matc)o(h)e(\(3.50\))i(on)f(the)g Fk(B)i Fr(lev)o(el,)c(w)o(e)h (need)122 80 y Fn(g)145 87 y Fm(0)165 80 y Fr(\()p Fn(y)r(;)8 b(s)p Fr(\))13 b Fk(!)h Fr(0)p Fn(;)22 b Fr(as)14 b Fn(s)g Fk(!)g(\0001)p Fn(:)0 182 y Fr(This)i(forces)g(us)h(to)g(tak)o(e)122 284 y Fn(C)157 291 y Fm(1)177 284 y Fr(\()p Fn(y)r Fr(\))c Fk(\021)g Fr(0)p Fn(;)1481 b Fr(\(3.66\))0 385 y(b)q(ecause)20 b(the)f(second)h(comp)q(onen)o(t)f(of)h(the)g(\014rst)g(term)e(on)i (the)f(righ)o(t)h(hand)g(side)g(of)g(\(3.64\))g(do)q(es)h(not)0 445 y(tend)16 b(to)h(zero)f(as)h Fn(s)c Fk(!)h(\0001)p Fr(.)21 b(W)l(e)16 b(see)g(this)g(b)o(y)g(using)g(the)g(asymptotic)f (form)o(ula)g(1)i(of)f(9.246)h(of)g([6],)122 578 y Fn(D)162 585 y Fj(q)181 578 y Fr(\()p Fn(z)r Fr(\))30 b(=)g Fn(e)365 558 y Ff(\000)p Fj(z)409 546 y Fd(2)428 558 y Fj(=)p Fm(4)473 578 y Fn(z)498 558 y Fj(q)534 505 y Fi( )575 578 y Fr(1)11 b Fk(\000)g(O)709 505 y Fi( )751 516 y(\014)751 541 y(\014)751 566 y(\014)751 591 y(\014)770 545 y Fn(q)p 769 567 25 2 v 769 612 a(z)799 516 y Fi(\014)799 541 y(\014)799 566 y(\014)799 591 y(\014)813 529 y Fm(2)841 505 y Fi(!)d(!)924 578 y Fn(;)105 b Fr(for)h(arg)18 b Fn(z)e(<)1388 545 y Fr(3)p Fn(\031)p 1388 567 54 2 v 1402 612 a Fr(4)1446 578 y Fn(;)365 b Fr(\(3.67\))0 717 y(and)17 b(noting)g(that)f Fk(k)p Fn(y)r Fk(k)p Fn(=s)e Fr(=)g Fk(O)q Fr(\()p Fn(\017)621 699 y Fj(\030)q Ff(\000)p Fj(\016)682 688 y Fe(0)695 717 y Fr(\))f Fk(!)h Fr(0)j(due)f(to)h(the)f (cuto\013.)73 778 y(The)h(second)g(comp)q(onen)o(t)e(of)i(the)g(second) f(term)f(on)i(the)g(righ)o(t)f(hand)h(side)g(of)f(\(3.64\))i(do)q(es)f (tend)g(to)0 838 y(zero)h(as)i Fn(s)e Fk(!)g(\0001)p Fr(.)28 b(F)l(orm)o(ula)17 b(\(3.67\))j(and)f(some)f(simple)e (estimates)h(sho)o(w)j(that)f(on)g(the)g(supp)q(ort)h(of)0 898 y Fn(F)7 b Fr(\()p Fk(k)p Fn(y)r Fk(k)p Fn(\017)154 880 y Fj(\016)171 868 y Fe(0)183 898 y Fr(\),)122 1043 y Fn(D)162 1050 y Fj(p)180 1056 y Fh(l)192 1050 y Ff(\000)p Fm(1)248 958 y Fi(0)248 1032 y(@)289 1010 y Fr(\()p Fk(\000)p Fr(1)k Fk(\000)g Fn(i)p Fr(\))p 289 1032 179 2 v 315 1040 a Fi(q)p 356 1040 87 2 v 356 1091 a Fn(b)377 1098 y Fm(1)397 1091 y Fn(\021)423 1074 y Fm(0)421 1102 y(1)473 1043 y Fr(\()p Fn(b)513 1050 y Fm(1)532 1043 y Fn(\021)558 1023 y Fm(0)556 1056 y(1)578 1043 y Fn(s)g Fr(+)g Fn(b)682 1050 y Fm(1)701 1043 y Fn(y)725 1050 y Fm(1)756 1043 y Fr(+)g Fn(b)826 1050 y Fm(2)845 1043 y Fn(y)869 1050 y Fm(2)889 1043 y Fr(\))908 958 y Fi(1)908 1032 y(A)985 1043 y Fr(=)42 b Fk(O)q Fr(\()p Fk(j)p Fn(s)p Fk(j)1176 1023 y Ff(\000)p Fm(1)1223 1043 y Fr(\))p Fn(:)0 1196 y Fr(Th)o(us,)20 b(the)f(\014rst)h(comp)q(onen)o(t)f(of)g(the)h(second) f(term)f(on)i(the)f(righ)o(t)g(hand)h(side)g(of)f(\(3.64\))h (determines)0 1256 y Fn(C)35 1263 y Fm(2)55 1256 y Fr(\()p Fn(y)r Fr(\))d(b)o(y)h(matc)o(hing)e(\(3.50\).)27 b(As)18 b Fn(s)f Fk(!)f(\0001)p Fr(,)i(that)g(comp)q(onen)o(t)f(has)i(the)f (follo)o(wing)g(asymptotics)f(b)o(y)0 1316 y(\(3.67\):)205 1462 y Fn(D)245 1469 y Fj(p)263 1475 y Fh(l)286 1376 y Fi(0)286 1451 y(@)327 1428 y Fr(\()p Fk(\000)p Fr(1)11 b Fk(\000)g Fn(i)p Fr(\))p 327 1450 179 2 v 353 1458 a Fi(q)p 394 1458 87 2 v 394 1510 a Fn(b)415 1517 y Fm(1)434 1510 y Fn(\021)460 1493 y Fm(0)458 1521 y(1)511 1462 y Fr(\()p Fn(b)551 1469 y Fm(1)570 1462 y Fn(\021)596 1441 y Fm(0)594 1474 y(1)616 1462 y Fn(s)g Fr(+)g Fn(b)720 1469 y Fm(1)739 1462 y Fn(y)763 1469 y Fm(1)794 1462 y Fr(+)g Fn(b)864 1469 y Fm(2)883 1462 y Fn(y)907 1469 y Fm(2)927 1462 y Fr(\))946 1376 y Fi(1)946 1451 y(A)205 1623 y Fr(=)30 b(exp)364 1550 y Fi( )405 1623 y Fk(\000)p Fn(i)466 1589 y Fr(\()p Fn(b)506 1596 y Fm(1)525 1589 y Fn(\021)551 1571 y Fm(0)549 1601 y(1)571 1589 y Fn(s)11 b Fr(+)g Fn(b)f Fk(\001)h Fn(y)r Fr(\))755 1571 y Fm(2)p 466 1611 310 2 v 565 1657 a Fr(2)p 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Fm(1)1238 1768 y Fn(\021)1264 1747 y Fm(0)1262 1780 y(1)1284 1768 y Fk(j)p Fn(s)p Fk(j)j(\000)f Fn(b)h Fk(\001)g Fn(y)r Fr(\))1505 1695 y Fi(#)e(!)254 1884 y Fk(\002)317 1836 y Fi(\020)358 1884 y Fr(1)20 b(+)f Fk(O)q Fr(\()p Fk(j)p Fn(s)p Fk(j)570 1863 y Ff(\000)p Fm(2)617 1884 y Fr(\))652 1836 y Fi(\021)685 1884 y Fn(:)0 1992 y Fr(In)12 b(the)g(matc)o(hing)e(regime,)g Fk(k)p Fn(y)r Fk(k)p Fn(=)p Fk(j)p Fn(s)p Fk(j)k(!)f Fr(0,)g(so)g(b)o(y)e(a)i (fairly)e(length)o(y)g(calculation,)h(this)g(quan)o(tit)o(y)f(matc)o (hes)0 2052 y(\(3.50\))17 b(if)f(w)o(e)f(c)o(ho)q(ose)i Fn(\027)436 2059 y Fm(0)456 2052 y Fr(\()p Fn(\017)p Fr(\))c(=)h(1)j(and)g Fn(C)750 2059 y Fm(2)769 2052 y Fr(\()p Fn(y)r Fr(\))f(to)h(b)q(e)18 2185 y Fn(C)53 2192 y Fm(2)73 2185 y Fr(\()p Fn(y)r Fr(\))41 b(=)g Fk(\000)8 b Fn(\017)324 2164 y Ff(\000)p Fj(n=)p Fm(2)418 2185 y Fn(')450 2192 y Fj(l)463 2185 y Fr(\()p Fn(A)519 2192 y Fm(0)539 2185 y Fn(;)g(B)598 2192 y Fm(0)617 2185 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p 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Fm(2)1253 2305 y(3)p 1195 2314 V 1195 2360 a Fr(2)p Fn(b)1240 2367 y Fm(1)1260 2360 y Fn(\021)1286 2343 y Fm(0)1284 2371 y(1)1327 2326 y Fr(ln)16 b Fk(j)p Fn(d)1423 2333 y Fm(3)1443 2326 y Fk(j)j Fr(+)11 b Fn(i)1567 2292 y Fr(3)p Fn(d)1616 2274 y Fm(2)1616 2305 y(3)p 1547 2314 V 1547 2360 a Fr(4)p Fn(b)1592 2367 y Fm(1)1612 2360 y Fn(\021)1638 2343 y Fm(0)1636 2371 y(1)1679 2326 y Fr(ln)d(\(2)p Fn(b)1792 2333 y Fm(1)1812 2326 y Fn(\021)1838 2306 y Fm(0)1836 2338 y(1)1857 2326 y Fr(\))1876 2253 y Fi(!)257 2467 y Fk(\002)25 b Fr(exp)403 2394 y Fi( )445 2467 y Fk(\000)p Fn(i)517 2434 y(c)538 2415 y Fm(2)538 2446 y(2)558 2434 y Fn(y)584 2415 y Fm(2)582 2446 y(2)p 506 2456 V 506 2501 a Fr(4)p Fn(b)551 2508 y Fm(1)570 2501 y Fn(\021)596 2484 y Fm(0)594 2512 y(1)637 2467 y Fr(ln)8 b(\(2)p Fn(b)750 2474 y Fm(1)770 2467 y Fn(\021)796 2447 y Fm(0)794 2480 y(1)815 2467 y Fr(\))j(+)g Fn(i)928 2434 y(c)949 2415 y Fm(2)949 2446 y(2)969 2434 y Fn(y)995 2415 y Fm(2)993 2446 y(2)p 916 2456 V 916 2501 a Fr(2)p Fn(b)961 2508 y Fm(1)981 2501 y Fn(\021)1007 2484 y Fm(0)1005 2512 y(1)1048 2467 y Fr(ln)16 b Fk(j)p Fn(d)1144 2474 y Fm(3)1164 2467 y Fk(j)j(\000)11 b Fn(i)1269 2434 y Fk(j)p Fn(d)1308 2441 y Fm(3)1328 2434 y Fk(j)p Fn(b)f Fk(\001)h Fn(y)p 1269 2456 156 2 v 1303 2501 a(b)1324 2508 y Fm(1)1344 2501 y Fn(\021)1370 2484 y Fm(0)1368 2512 y(1)1437 2394 y Fi(!)1478 2467 y Fn(:)333 b Fr(\(3.68\))73 2597 y(The)16 b(follo)o(wing)g(result)g(deals)g(with)g(the)g(v)m (alidit)o(y)f(of)i(the)f(inner)f(appro)o(ximate)g(solution.)951 2753 y(35)p eop %%Page: 36 36 36 35 bop 0 -22 a Fg(Lemma)16 b(3.5)24 b Fl(Supp)n(ose)c Fr(0)f Fn(<)h(\016)608 -40 y Ff(0)638 -22 y Fn(<)f(\030)j(<)d Fr(1)p Fn(=)p Fr(3)p Fl(,)j Fn(\024)d(>)1013 -41 y Fm(1)p 1013 -33 18 2 v 1013 -4 a(2)1036 -22 y Fl(,)i(and)f Fr(1)14 b Fk(\000)f Fn(\030)22 b(>)d(\024)p Fl(.)31 b(The)20 b(function)i Fr(\011)1768 -15 y Fj(I)1788 -22 y Fr(\()p Fn(x;)8 b(t)p Fr(\))18 b(=)0 39 y Fn( )32 46 y Fj(I)52 39 y Fr(\(\()p Fn(x)t Fk(\000)t Fn(a)p Fr(\()p Fn(t)p Fr(\)\))p Fn(=\017;)8 b(t=\017)p Fr(\))k Fl(is)j(a)f(valid)h(appr)n (oximation)e(of)h(a)h(solution)g Fn( )r Fr(\()p Fn(x;)8 b(t)p Fr(\))13 b Fl(to)h(the)h(Schr\177)-25 b(odinger)15 b(e)n(quation)0 99 y(for)i Fk(\000)p Fn(\017)137 81 y Fm(1)p Ff(\000)p Fj(\030)214 99 y Fk(\024)d Fn(t)f Fk(\024)h Fn(\017)371 81 y Fm(1)p Ff(\000)p Fj(\030)434 99 y Fl(,)k(in)g(the)f (sense)i(that)f(as)f Fn(\017)c Fk(!)h Fr(0)p Fl(,)122 193 y Fk(k)8 b Fn( )r Fr(\()p Fk(\001)p Fn(;)g(t)p Fr(\))i Fk(\000)h Fr(\011)379 200 y Fj(I)399 193 y Fr(\()p Fk(\001)p Fn(;)d(t)p Fr(\))g Fk(k)13 b Fr(=)h Fk(O)q Fr(\()p Fn(\017)669 173 y Fm(1)p Ff(\000)p Fm(3)p Fj(\030)750 193 y Fr(\))g Fk(!)f Fr(0)p Fn(:)0 288 y Fl(F)l(urthermor)n(e,)k(in)h(the)g(matching) g(r)n(e)n(gime,)f Fk(\000)p Fn(\017)877 269 y Fj(\024)913 288 y Fn(<)d(t)g(<)g Fk(\000)p Fn(\017)1108 269 y Fm(1)p Ff(\000)p Fj(\030)1171 288 y Fl(,)k(the)g(inner)g(and)g(outer)g (solutions)g(agr)n(e)n(e)0 348 y(in)g(the)g(sense)g(that)122 442 y Fk(k)8 b Fr(\011)193 449 y Fj(I)s(O)241 442 y Fr(\()p Fk(\001)p Fn(;)g(t)p Fr(\))i Fk(\000)h Fr(\011)431 449 y Fj(I)451 442 y Fr(\()p Fk(\001)p Fn(;)d(t)p Fr(\))g Fk(k)13 b Fr(=)h Fk(O)q Fr(\()p Fn(\017)721 422 y Fj(\013)745 442 y Fr(\))0 537 y Fl(for)j(some)g Fn(\013)e(>)e Fr(0)p Fl(.)0 641 y Fg(Pro)r(of:)0 701 y Fr(The)k(\014rst)h(result)f(is)g(pro) o(v)o(ed)g(b)o(y)g(mimi)o(c)o(ki)o(ng)e(the)i(pro)q(ofs)i(of)f(prop)q (osition)g(3.1)g(of)g([20])f(and)h(lemma)d(3.5)0 761 y(of)h([20)q(].)k(The)c(second)g(result)g(is)g(pro)o(v)o(ed)f(b)o(y)g (com)o(bining)f(lemma)f(3.4)j(and)h(simple)d(estimates)g(based)j(on)0 821 y(the)f(calculations)g(presen)o(ted)f(ab)q(o)o(v)o(e.)1209 b Fa(2)73 972 y Fr(W)l(e)14 b(need)g(the)g(large)h(p)q(ositiv)o(e)e Fn(s)h Fr(asymptotics)g(of)g(the)g(inner)g(solution)h(to)g(matc)o(h)d (it)i(to)h(an)g(outgoing)0 1032 y(outer)h(solution.)22 b(T)l(o)17 b(obtain)f(these)g(asymptotics,)f(w)o(e)h(use)g(form)o(ula)f (3)i(of)f(9.246)i(of)e([6],)122 1162 y Fn(D)162 1169 y Fj(q)181 1162 y Fr(\()p Fn(z)r Fr(\))42 b(=)f Fn(e)388 1141 y Ff(\000)p Fj(z)432 1129 y Fd(2)450 1141 y Fj(=)p Fm(4)496 1162 y Fn(z)521 1141 y Fj(q)557 1088 y Fi( )598 1162 y Fr(1)11 b Fk(\000)g(O)732 1088 y Fi( )773 1099 y(\014)773 1124 y(\014)773 1149 y(\014)773 1174 y(\014)793 1128 y Fn(q)p 792 1150 25 2 v 792 1196 a(z)822 1099 y Fi(\014)822 1124 y(\014)822 1149 y(\014)822 1174 y(\014)836 1112 y Fm(2)864 1088 y Fi(!)d(!)949 1162 y Fk(\000)1022 1087 y(p)p 1063 1087 54 2 v 1063 1128 a Fr(2)p Fn(\031)p 1004 1150 131 2 v 1004 1196 a Fr(\000\()p Fk(\000)p Fn(q)r Fr(\))1148 1162 y Fn(e)1171 1141 y Ff(\000)p Fj(q)q(\031)q(i)1259 1162 y Fn(e)1282 1141 y Fj(z)1299 1129 y Fd(2)1316 1141 y Fj(=)p Fm(4)1362 1162 y Fn(z)1387 1141 y Ff(\000)p Fj(q)q Ff(\000)p Fm(1)1495 1088 y Fi( )1536 1162 y Fr(1)k Fk(\000)e(O)1671 1088 y Fi( )1712 1099 y(\014)1712 1124 y(\014)1712 1149 y(\014)1712 1174 y(\014)1732 1128 y Fn(q)p 1731 1150 25 2 v 1731 1196 a(z)1761 1099 y Fi(\014)1761 1124 y(\014)1761 1149 y(\014)1761 1174 y(\014)1774 1112 y Fm(2)1803 1088 y Fi(!)e(!)1885 1162 y Fn(;)0 1291 y Fr(for)98 b Fk(\000)200 1271 y Fm(5)p Fj(\031)p 200 1279 40 2 v 210 1308 a Fm(4)257 1291 y Fn(<)14 b Fr(arg)k Fn(z)e(<)490 1271 y Fj(\031)p 490 1279 22 2 v 492 1308 a Fm(4)516 1291 y Fr(.)73 1351 y(F)l(rom)g(this)i(form)o(ula,)d(the)j (\014rst)f(comp)q(onen)o(t)g(of)g(the)g(second)h(term)e(in)h(\(3.64\))h (has)g(large)f Fn(s)h Fr(asymp-)0 1411 y(totics)e(\(omitting)f(the)h (factor)g Fn(C)608 1418 y Fm(2)628 1411 y Fr(\()p Fn(y)r Fr(\)\))205 1549 y Fn(D)245 1556 y Fj(p)263 1562 y Fh(l)286 1464 y Fi(0)286 1538 y(@)327 1515 y Fr(\()p Fk(\000)p Fr(1)11 b Fk(\000)g Fn(i)p Fr(\))p 327 1537 179 2 v 353 1546 a Fi(q)p 394 1546 87 2 v 394 1597 a Fn(b)415 1604 y Fm(1)434 1597 y Fn(\021)460 1580 y Fm(0)458 1608 y(1)511 1549 y Fr(\()p Fn(b)551 1556 y Fm(1)570 1549 y Fn(\021)596 1528 y Fm(0)594 1561 y(1)616 1549 y Fn(s)g Fr(+)g Fn(b)720 1556 y Fm(1)739 1549 y Fn(y)763 1556 y Fm(1)794 1549 y Fr(+)g Fn(b)864 1556 y Fm(2)883 1549 y Fn(y)907 1556 y Fm(2)927 1549 y Fr(\))946 1464 y Fi(1)946 1538 y(A)205 1710 y Fr(=)30 b(exp)364 1637 y Fi( )405 1710 y Fk(\000)p Fn(i)466 1676 y Fr(\()p Fn(b)506 1683 y Fm(1)525 1676 y Fn(\021)551 1658 y Fm(0)549 1689 y(1)571 1676 y Fn(s)11 b Fr(+)g Fn(b)f Fk(\001)h Fn(y)r Fr(\))755 1658 y Fm(2)p 466 1698 310 2 v 565 1744 a Fr(2)p Fn(b)610 1751 y Fm(1)630 1744 y Fn(\021)656 1727 y Fm(0)654 1755 y(1)799 1710 y Fk(\000)862 1676 y Fr(3)p Fn(\031)r Fr(\()p Fn(c)956 1658 y Fm(2)956 1689 y(2)976 1676 y Fn(y)1002 1658 y Fm(2)1000 1689 y(2)1032 1676 y Fr(+)g Fn(d)1106 1658 y Fm(2)1106 1689 y(3)1126 1676 y Fr(\))p 862 1698 283 2 v 948 1744 a(8)p Fn(b)993 1751 y Fm(1)1013 1744 y Fn(\021)1039 1727 y Fm(0)1037 1755 y(1)1158 1637 y Fi(!)400 1855 y Fk(\002)24 b Fr(exp)554 1782 y Fi( )596 1855 y Fk(\000)p Fn(i)657 1821 y Fr(\()p Fn(c)697 1803 y Fm(2)697 1834 y(2)716 1821 y Fn(y)742 1803 y Fm(2)740 1834 y(2)772 1821 y Fr(+)11 b Fn(d)846 1803 y Fm(2)846 1834 y(3)866 1821 y Fr(\))p 657 1844 230 2 v 715 1889 a(2)p Fn(b)760 1896 y Fm(1)780 1889 y Fn(\021)806 1872 y Fm(0)804 1900 y(1)915 1855 y Fr(ln)972 1782 y Fi( )1005 1780 y(s)p 1046 1780 96 2 v 1082 1821 a Fr(2)p 1051 1844 87 2 v 1051 1889 a Fn(b)1072 1896 y Fm(1)1092 1889 y Fn(\021)1118 1872 y Fm(0)1116 1900 y(1)1158 1855 y Fr(\()p Fn(b)1198 1862 y Fm(1)1218 1855 y Fn(\021)1244 1835 y Fm(0)1242 1867 y(1)1263 1855 y Fn(s)h Fr(+)f Fn(b)f Fk(\001)h Fn(y)r Fr(\))1456 1782 y Fi(!)d(!)400 1971 y Fk(\002)463 1923 y Fi(\020)504 1971 y Fr(1)20 b(+)f Fk(O)q Fr(\()p Fk(j)p Fn(s)p Fk(j)716 1951 y Ff(\000)p Fm(2)763 1971 y Fr(\))799 1923 y Fi(\021)254 2063 y Fr(+)d Fk(O)357 2015 y Fi(\020)390 2063 y Fk(j)p Fn(s)p Fk(j)441 2042 y Ff(\000)p Fm(1)496 2015 y Fi(\021)530 2063 y Fn(:)1281 b Fr(\(3.69\))0 2163 y(The)15 b(second)h(comp)q(onen)o(t)e(of)h(the)g(second)h(term)d(in)i (\(3.64\))h(has)g(large)f Fn(s)g Fr(asymptotics)f(\(again)i(omitting)0 2224 y(the)g(factor)h Fn(C)259 2231 y Fm(2)278 2224 y Fr(\()p Fn(y)r Fr(\)\))210 2328 y(\()p Fk(\000)p Fr(1)11 b Fk(\000)g Fn(i)p Fr(\)\()p Fn(c)429 2335 y Fm(2)448 2328 y Fn(y)472 2335 y Fm(2)503 2328 y Fk(\000)g Fn(id)595 2335 y Fm(3)614 2328 y Fr(\))p 210 2350 424 2 v 346 2410 a(2)370 2358 y Fi(q)p 412 2358 87 2 v 52 x Fn(b)433 2417 y Fm(1)452 2410 y Fn(\021)478 2392 y Fm(0)476 2420 y(1)647 2361 y Fn(D)687 2368 y Fj(p)705 2374 y Fh(l)717 2368 y Ff(\000)p Fm(1)772 2276 y Fi(0)772 2351 y(@)814 2328 y Fr(\()p Fk(\000)p Fr(1)g Fk(\000)g Fn(i)p Fr(\))p 814 2350 179 2 v 839 2358 a Fi(q)p 881 2358 87 2 v 52 x Fn(b)902 2417 y Fm(1)921 2410 y Fn(\021)947 2392 y Fm(0)945 2420 y(1)997 2361 y Fr(\()p Fn(b)1037 2368 y Fm(1)1057 2361 y Fn(\021)1083 2341 y Fm(0)1081 2374 y(1)1102 2361 y Fn(s)g Fr(+)g Fn(b)1206 2368 y Fm(1)1226 2361 y Fn(y)1250 2368 y Fm(1)1280 2361 y Fr(+)g Fn(b)1350 2368 y Fm(2)1370 2361 y Fn(y)1394 2368 y Fm(2)1413 2361 y Fr(\))1432 2276 y Fi(1)1432 2351 y(A)205 2569 y Fr(=)278 2536 y(\(1)g(+)g Fn(i)p Fr(\)\()p Fn(c)457 2543 y Fm(2)477 2536 y Fn(y)501 2543 y Fm(2)531 2536 y Fk(\000)g Fn(id)623 2543 y Fm(3)643 2536 y Fr(\))p 278 2558 384 2 v 394 2617 a(2)418 2566 y Fi(q)p 460 2566 87 2 v 51 x Fn(b)481 2624 y Fm(1)500 2617 y Fn(\021)526 2600 y Fm(0)524 2628 y(1)720 2495 y Fk(p)p 762 2495 54 2 v 41 x Fr(2)p Fn(\031)f(e)847 2518 y Fj(i\031)891 2536 y Fn(e)914 2497 y Ff(\000)946 2474 y Fh(\031)q Fd(\()p Fh(c)991 2464 y Fd(2)991 2487 y(2)1008 2474 y Fh(y)1025 2464 y Fd(2)1024 2487 y(2)1042 2474 y(+)p Fh(d)1081 2464 y Fd(2)1081 2487 y(3)1098 2474 y(\))p 946 2490 165 2 v 988 2516 a(2)p Fh(b)1017 2523 y Fd(1)1034 2516 y Fh(\021)1051 2508 y Fd(0)1050 2530 y(1)p 720 2558 397 2 v 740 2616 a Fr(\000)778 2568 y Fi(\020)803 2616 y Fr(1)i(+)f Fn(i)910 2590 y Fm(\()p Fj(c)939 2579 y Fd(2)939 2601 y(2)956 2590 y Fj(y)974 2579 y Fd(2)973 2601 y(2)992 2590 y Fm(+)p Fj(d)1037 2579 y Fd(2)1037 2601 y(3)1055 2590 y Fm(\))p 910 2605 159 2 v 946 2634 a(2)p Fj(b)979 2639 y Fd(1)996 2634 y Fj(\021)1015 2623 y Fd(0)1014 2645 y(1)1073 2568 y Fi(\021)951 2753 y Fr(36)p eop %%Page: 37 37 37 36 bop 400 3 a Fk(\002)24 b Fr(exp)554 -70 y Fi( )596 3 y Fn(i)618 -30 y Fr(\()p Fn(b)658 -23 y Fm(1)677 -30 y Fn(\021)703 -48 y Fm(0)701 -18 y(1)722 -30 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))907 -48 y Fm(2)p 618 -8 310 2 v 717 37 a Fr(2)p Fn(b)762 44 y Fm(1)781 37 y Fn(\021)807 20 y Fm(0)805 48 y(1)951 3 y Fr(+)1013 -30 y(3)p Fn(\031)r Fr(\()p Fn(c)1107 -48 y Fm(2)1107 -18 y(2)1126 -30 y Fn(y)1152 -48 y Fm(2)1150 -18 y(2)1183 -30 y Fr(+)g Fn(d)1257 -48 y Fm(2)1257 -18 y(3)1277 -30 y Fr(\))p 1013 -8 283 2 v 1099 37 a(8)p Fn(b)1144 44 y Fm(1)1164 37 y Fn(\021)1190 20 y Fm(0)1188 48 y(1)1309 -70 y Fi(!)400 148 y Fk(\002)24 b Fr(exp)554 75 y Fi( )596 148 y Fn(i)618 115 y Fr(\()p Fn(c)658 97 y Fm(2)658 127 y(2)677 115 y Fn(y)703 97 y Fm(2)701 127 y(2)733 115 y Fr(+)11 b Fn(d)807 97 y Fm(2)807 127 y(3)827 115 y Fr(\))p 618 137 230 2 v 677 183 a(2)p Fn(b)722 190 y Fm(1)741 183 y Fn(\021)767 165 y Fm(0)765 193 y(1)868 148 y Fr(ln)925 75 y Fi( )958 73 y(s)p 1000 73 96 2 v 1035 115 a Fr(2)p 1005 137 87 2 v 1005 183 a Fn(b)1026 190 y Fm(1)1045 183 y Fn(\021)1071 165 y Fm(0)1069 193 y(1)1112 148 y Fr(\()p Fn(b)1152 155 y Fm(1)1171 148 y Fn(\021)1197 128 y Fm(0)1195 161 y(1)1217 148 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1410 75 y Fi(!)d(!)400 265 y Fk(\002)463 216 y Fi(\020)504 265 y Fr(1)20 b(+)f Fk(O)q Fr(\()p Fk(j)p Fn(s)p Fk(j)716 244 y Ff(\000)p Fm(2)763 265 y Fr(\))799 216 y Fi(\021)254 356 y Fr(+)d Fk(O)357 308 y Fi(\020)390 356 y Fk(j)p Fn(s)p Fk(j)441 335 y Ff(\000)p Fm(1)496 308 y Fi(\021)205 471 y Fr(=)278 437 y(\()p Fk(\000)p Fr(1)11 b Fk(\000)g Fn(i)p Fr(\)\()p Fn(c)497 444 y Fm(2)516 437 y Fn(y)540 444 y Fm(2)571 437 y Fk(\000)g Fn(id)663 444 y Fm(3)682 437 y Fr(\))p 278 459 424 2 v 414 467 a Fi(q)p 455 467 111 2 v 455 519 a Fr(2)p Fn(b)500 526 y Fm(1)520 519 y Fn(\021)546 502 y Fm(0)544 530 y(1)904 401 y Fk(p)p 945 401 30 2 v 945 437 a Fn(\031)p 760 459 359 2 v 760 518 a Fr(\000)798 470 y Fi(\020)824 518 y Fr(1)g(+)g Fn(i)930 492 y Fm(\()p Fj(c)959 480 y Fd(2)959 503 y(2)976 492 y Fj(y)994 480 y Fd(2)993 503 y(2)1012 492 y Fm(+)p Fj(d)1057 480 y Fd(2)1057 503 y(3)1075 492 y Fm(\))p 930 506 159 2 v 966 535 a(2)p Fj(b)999 540 y Fd(1)1016 535 y Fj(\021)1035 524 y Fd(0)1034 547 y(1)1094 470 y Fi(\021)400 638 y Fk(\002)24 b Fr(exp)554 565 y Fi( )596 638 y Fn(i)618 604 y Fr(\()p Fn(b)658 611 y Fm(1)677 604 y Fn(\021)703 586 y Fm(0)701 617 y(1)722 604 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))907 586 y Fm(2)p 618 626 310 2 v 717 672 a Fr(2)p Fn(b)762 679 y Fm(1)781 672 y Fn(\021)807 655 y Fm(0)805 683 y(1)951 638 y Fk(\000)1014 604 y Fn(\031)r Fr(\()p Fn(c)1084 586 y Fm(2)1084 617 y(2)1103 604 y Fn(y)1129 586 y Fm(2)1127 617 y(2)1159 604 y Fr(+)g Fn(d)1233 586 y Fm(2)1233 617 y(3)1253 604 y Fr(\))p 1014 626 259 2 v 1088 672 a(8)p Fn(b)1133 679 y Fm(1)1153 672 y Fn(\021)1179 655 y Fm(0)1177 683 y(1)1286 565 y Fi(!)400 783 y Fk(\002)24 b Fr(exp)554 710 y Fi( )596 783 y Fn(i)618 749 y Fr(\()p Fn(c)658 731 y Fm(2)658 762 y(2)677 749 y Fn(y)703 731 y Fm(2)701 762 y(2)733 749 y Fr(+)11 b Fn(d)807 731 y Fm(2)807 762 y(3)827 749 y Fr(\))p 618 771 230 2 v 677 817 a(2)p Fn(b)722 824 y Fm(1)741 817 y Fn(\021)767 800 y Fm(0)765 828 y(1)868 783 y Fr(ln)925 710 y Fi( )958 708 y(s)p 1000 708 96 2 v 1035 749 a Fr(2)p 1005 771 87 2 v 1005 817 a Fn(b)1026 824 y Fm(1)1045 817 y Fn(\021)1071 800 y Fm(0)1069 828 y(1)1112 783 y Fr(\()p Fn(b)1152 790 y Fm(1)1171 783 y Fn(\021)1197 762 y Fm(0)1195 795 y(1)1217 783 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1410 710 y Fi(!)d(!)400 899 y Fk(\002)463 851 y Fi(\020)504 899 y Fr(1)20 b(+)f Fk(O)q Fr(\()p Fk(j)p Fn(s)p Fk(j)716 879 y Ff(\000)p Fm(2)763 899 y Fr(\))799 851 y Fi(\021)254 991 y Fr(+)d Fk(O)357 942 y Fi(\020)390 991 y Fk(j)p Fn(s)p Fk(j)441 970 y Ff(\000)p Fm(1)496 942 y Fi(\021)530 991 y Fn(:)1281 b Fr(\(3.70\))0 1176 y Fc(3.5)70 b(Outgoing)22 b(Outer)h(Solution)0 1269 y Fr(T)l(o)17 b(construct)f(the)g(outer)h(solution)f(for)h(p)q(ositiv)o (e)e(times,)f(w)o(e)i(use)g(the)g(classical)g(quan)o(tities)f(asso)q (ciated)0 1329 y(with)h(the)g Fk(A)g Fr(and)h Fk(B)g Fr(lev)o(els)e(that)h(satisfy)h(the)f(follo)o(wing)g(initial)f (conditions:)29 1431 y Fn(a)55 1410 y Ff(A)85 1431 y Fr(\(0\))42 b(=)g(0)p Fn(;)1518 b Fr(\(3.71\))29 1531 y Fn(\021)55 1510 y Ff(A)85 1531 y Fr(\(0\))42 b(=)g Fn(\021)295 1510 y Fm(0)334 1531 y Fk(\000)397 1497 y Fr(2)p Fk(j)p Fn(d)460 1504 y Fm(3)480 1497 y Fk(j)p Fn(\017)p 397 1519 117 2 v 413 1565 a(b)434 1572 y Fm(1)453 1565 y Fn(\021)479 1548 y Fm(0)477 1576 y(1)535 1531 y Fn(b;)22 1636 y(S)55 1615 y Ff(A)85 1636 y Fr(\(0\))g(=)g(0)p Fn(;)18 1751 y(A)55 1731 y Ff(A)85 1751 y Fr(\(0\))g(=)269 1678 y Fi( )310 1751 y Fr(\()p Fn(A)366 1758 y Fm(0)386 1751 y Fn(A)423 1731 y Ff(\003)423 1764 y Fm(0)442 1751 y Fr(\))461 1731 y Ff(\000)p Fm(1)527 1751 y Fr(+)618 1718 y Fn(\031)p 590 1740 87 2 v 590 1785 a(b)611 1792 y Fm(1)630 1785 y Fn(\021)656 1768 y Fm(0)654 1796 y(1)689 1751 y Fk(j)p Fn(c)p Fk(ih)p Fn(c)p Fk(j)805 1678 y Fi(!)838 1690 y Ff(\000)p Fm(1)p Fj(=)p Fm(2)22 1892 y Fn(B)62 1872 y Ff(A)92 1892 y Fr(\()p Fn(t)p Fr(\))f Fk(\030)269 1819 y Fi( )310 1892 y Fn(B)347 1899 y Fm(0)367 1892 y Fn(A)404 1872 y Ff(\000)p Fm(1)404 1905 y(0)470 1892 y Fr(+)561 1859 y Fn(\031)p 532 1881 V 532 1927 a(b)553 1934 y Fm(1)573 1927 y Fn(\021)599 1909 y Fm(0)597 1938 y(1)632 1892 y Fk(j)p Fn(c)p Fk(ih)p Fn(c)p Fk(j)748 1819 y Fi(!)798 1892 y Fn(A)835 1872 y Ff(A)865 1892 y Fr(\(0\))19 b(+)1009 1859 y Fn(i)p Fk(j)p Fn(d)1065 1866 y Fm(3)1084 1859 y Fk(j)p 1009 1881 90 2 v 1011 1927 a Fn(b)1032 1934 y Fm(1)1051 1927 y Fn(\021)1077 1909 y Fm(0)1075 1938 y(1)1111 1819 y Fi(")1136 1892 y Fn(M)5 b(A)1225 1872 y Ff(A)1255 1892 y Fr(\(0\))20 b(+)f(ln)1451 1819 y Fi( )1489 1859 y Fr(2)p Fn(b)1534 1866 y Fm(1)1554 1859 y Fn(\021)1580 1841 y Fm(0)1578 1871 y(1)1599 1859 y Fn(t)p 1489 1881 129 2 v 1507 1927 a Fk(j)p Fn(d)1546 1934 y Fm(3)1566 1927 y Fk(j)p Fn(\017)1622 1819 y Fi(!)1663 1892 y Fn(N)5 b(A)1744 1872 y Ff(A)1774 1892 y Fr(\(0\))1836 1819 y Fi(#)1869 1892 y Fn(;)33 2069 y(a)59 2048 y Ff(B)85 2069 y Fr(\(0\))42 b(=)g(0)p Fn(;)1518 b Fr(\(3.72\))33 2142 y Fn(\021)59 2121 y Ff(B)85 2142 y Fr(\(0\))42 b(=)g Fn(\021)295 2121 y Fm(0)315 2142 y Fn(;)26 2214 y(S)59 2194 y Ff(B)85 2214 y Fr(\(0\))g(=)g(0)p Fn(;)22 2287 y(A)59 2266 y Ff(B)85 2287 y Fr(\(0\))g(=)g Fn(A)306 2294 y Fm(0)325 2287 y Fn(;)26 2391 y(B)66 2370 y Ff(B)92 2391 y Fr(\()p Fn(t)p Fr(\))f Fk(\030)g Fn(B)306 2398 y Fm(0)345 2391 y Fk(\000)408 2357 y Fn(i)p Fk(j)p Fn(d)464 2364 y Fm(3)484 2357 y Fk(j)p 408 2379 90 2 v 410 2425 a Fn(b)431 2432 y Fm(1)450 2425 y Fn(\021)476 2408 y Fm(0)474 2436 y(1)519 2391 y Fr(ln)577 2318 y Fi( )647 2357 y Fn(d)672 2339 y Fm(2)672 2369 y(3)p 614 2379 111 2 v 614 2425 a Fr(2)p Fn(b)659 2432 y Fm(1)679 2425 y Fn(\021)705 2408 y Fm(0)703 2436 y(1)730 2318 y Fi(!)779 2391 y Fn(N)5 b(A)860 2398 y Fm(0)907 2391 y Fk(\000)978 2357 y Fn(i)p Fk(j)p Fn(d)1034 2364 y Fm(3)1054 2357 y Fk(j)p 978 2379 90 2 v 980 2425 a Fn(b)1001 2432 y Fm(1)1020 2425 y Fn(\021)1046 2408 y Fm(0)1044 2436 y(1)1081 2318 y Fi(")1105 2391 y Fn(M)g(A)1194 2398 y Fm(0)1233 2391 y Fr(+)19 b(ln)1348 2318 y Fi( )1385 2357 y Fr(2)p Fn(b)1430 2364 y Fm(1)1450 2357 y Fn(\021)1476 2339 y Fm(0)1474 2369 y(1)1496 2357 y Fn(t)p 1385 2379 129 2 v 1403 2425 a Fk(j)p Fn(d)1442 2432 y Fm(3)1462 2425 y Fk(j)p Fn(\017)1519 2318 y Fi(!)1560 2391 y Fn(N)5 b(A)1641 2398 y Fm(0)1660 2318 y Fi(#)1693 2391 y Fn(:)951 2753 y Fr(37)p eop %%Page: 38 38 38 37 bop 0 -22 a Fr(Here)18 b(the)h(asymptotic)g(conditions)g(for)h Fn(B)814 -40 y Ff(C)836 -22 y Fr(\()p Fn(t)p Fr(\))f(are)g(to)h(b)q(e)g (satis\014ed)f(in)g(the)h(regime)d(where)i Fn(t)p Fr(,)g Fn(\017)p Fr(,)h Fn(\017=t)p Fr(,)0 39 y(and)d Fn(t)8 b Fr(ln)161 17 y Fm(2)181 39 y Fr(\()p Fn(\017)p Fr(\))16 b(are)g(all)g(small.)k(It)15 b(is)i(con)o(v)o(enien)o(t)d(to)i (de\014ne)g(the)g(quan)o(tit)o(y)122 166 y Fn(B)162 146 y Ff(A)159 179 y Fm(1)192 166 y Fr(\()p Fn(t)p Fr(\))41 b(=)g Fn(B)405 173 y Fm(0)444 166 y Fr(+)507 133 y Fn(i)p Fk(j)p Fn(d)563 140 y Fm(3)582 133 y Fk(j)p 507 155 90 2 v 509 201 a Fn(b)530 208 y Fm(1)549 201 y Fn(\021)575 183 y Fm(0)573 211 y(1)609 93 y Fi(")634 166 y Fn(M)5 b(A)723 173 y Fm(0)762 166 y Fr(+)19 b(ln)876 93 y Fi( )914 133 y Fr(2)p Fn(b)959 140 y Fm(1)979 133 y Fn(\021)1005 115 y Fm(0)1003 145 y(1)1024 133 y Fn(t)p 914 155 129 2 v 932 201 a Fk(j)p Fn(d)971 208 y Fm(3)991 201 y Fk(j)p Fn(\017)1047 93 y Fi(!)1088 166 y Fn(N)5 b(A)1169 173 y Fm(0)1189 93 y Fi(#)1221 166 y Fn(:)73 291 y Fr(One)17 b(should)g(note)g(the)g(di\013erence)e(in)i(initial)e(momen)o(tum)e (with)k(the)f(classical)g(quan)o(tities)g(asso)q(ci-)0 352 y(ated)h(with)g(the)g Fk(B)h Fr(lev)o(el,)d(\(3.43\).)24 b(As)17 b(explained)f(earlier,)g(the)h(corresp)q(onding)h(loss)f(in)g (kinetic)e(energy)0 412 y(equals)h(the)g(gain)h(in)f(p)q(oten)o(tial)f (energy)l(,)h(to)g(leading)g(order.)73 472 y(One)i(should)g(also)g 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2640 y(1)585 2595 y Fr(ln)8 b(\(2)p Fn(b)698 2602 y Fm(1)717 2595 y Fn(\021)743 2575 y Fm(0)741 2608 y(1)763 2595 y Fr(\))j(+)g Fn(i)876 2562 y(c)897 2544 y Fm(2)897 2574 y(2)917 2562 y Fn(y)943 2544 y Fm(2)941 2574 y(2)p 864 2584 V 864 2630 a Fr(2)p Fn(b)909 2637 y Fm(1)928 2630 y Fn(\021)954 2612 y Fm(0)952 2640 y(1)996 2595 y Fr(ln)16 b Fk(j)p Fn(d)1092 2602 y Fm(3)1112 2595 y Fk(j)j(\000)11 b Fn(i)1217 2562 y Fk(j)p Fn(d)1256 2569 y Fm(3)1275 2562 y Fk(j)d Fn(b)j Fk(\001)g Fn(y)p 1217 2584 164 2 v 1255 2630 a(b)1276 2637 y Fm(1)1295 2630 y Fn(\021)1321 2612 y Fm(0)1319 2640 y(1)1393 2522 y Fi(!)951 2753 y Fr(38)p eop %%Page: 39 39 39 38 bop 205 3 a Fk(\002)24 b Fr(exp)359 -70 y Fi( )401 3 y Fk(\000)p Fn(i)462 -30 y(b)483 -23 y Fm(1)501 -30 y Fn(\021)527 -48 y Fm(0)525 -18 y(1)547 -30 y Fn(s)570 -48 y Fm(2)p 462 -8 129 2 v 513 37 a Fr(2)614 3 y Fk(\000)19 b Fn(is)8 b(b)j Fk(\001)g Fn(y)21 b Fk(\000)e Fn(i)902 -30 y Fr(\()p Fn(b)10 b Fk(\001)h Fn(y)r Fr(\))1022 -48 y Fm(2)p 902 -8 140 2 v 916 37 a Fr(2)p Fn(b)961 44 y Fm(1)981 37 y Fn(\021)1007 20 y Fm(0)1005 48 y(1)1066 3 y Fk(\000)1129 -30 y Fr(3)p Fn(\031)r Fr(\()p Fn(c)1223 -48 y Fm(2)1223 -18 y(2)1242 -30 y Fn(y)1268 -48 y Fm(2)1266 -18 y(2)1299 -30 y Fr(+)g Fn(d)1373 -48 y Fm(2)1373 -18 y(3)1393 -30 y Fr(\))p 1129 -8 283 2 v 1215 37 a(8)p Fn(b)1260 44 y Fm(1)1280 37 y Fn(\021)1306 20 y Fm(0)1304 48 y(1)1425 -70 y Fi(!)205 148 y Fk(\002)24 b Fr(exp)359 75 y Fi( )401 148 y Fk(\000)p Fn(i)462 115 y Fr(\()p Fn(c)502 97 y Fm(2)502 127 y(2)521 115 y Fn(y)547 97 y Fm(2)545 127 y(2)577 115 y Fr(+)11 b Fn(d)651 97 y Fm(2)651 127 y(3)671 115 y Fr(\))p 462 137 230 2 v 520 183 a(2)p Fn(b)565 190 y Fm(1)585 183 y Fn(\021)611 165 y Fm(0)609 193 y(1)720 148 y Fr(ln)777 75 y Fi( )818 73 y(s)p 859 73 96 2 v 895 115 a Fr(2)p 864 137 87 2 v 864 183 a Fn(b)885 190 y Fm(1)905 183 y Fn(\021)931 165 y Fm(0)929 193 y(1)972 148 y Fr(\()p Fn(b)1012 155 y Fm(1)1031 148 y Fn(\021)1057 128 y Fm(0)1055 161 y(1)1077 148 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)f Fn(y)r Fr(\))1269 75 y Fi(!)e(!)1352 148 y Fn(:)459 b Fr(\(3.74\))0 258 y(By)18 b(making)g(a)i(small)d(error,)j(w)o(e)f(can)g(drop)h(the)e Fn(b)13 b Fk(\001)g Fn(y)21 b Fr(term)c(in)i(the)g(logarithm)f(in)h (the)g(\014nal)g(term)e(in)0 318 y(this)f(expression.)21 b(Also,)16 b(b)o(y)g(prop)q(osition)h(2.2,)g(w)o(e)e(mak)o(e)g(a)i (small)d(error)j(b)o(y)f(replacing)f Fn(F)7 b Fr(\()p Fn(\017)1731 300 y Ff(\000)p Fj(\016)1775 288 y Fe(0)1788 318 y Fk(k)p Fn(y)r Fk(k)p Fr(\))16 b(b)o(y)0 378 y Fn(F)7 b Fr(\()p Fn(\017)78 360 y Ff(\000)p Fj(\016)122 348 y Fe(0)135 378 y Fk(k)p Fn(y)186 360 y Ff(A)215 378 y Fk(k)p Fr(\).)21 b(Th)o(us,)c(simple)d(calculations)h(sho)o(w)i(that)g (\(3.74\))g(appro)o(ximately)d(equals)139 490 y Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)264 469 y Fj(\016)281 458 y Fe(0)293 490 y Fk(k)p Fn(y)344 469 y Ff(A)374 490 y Fk(k)p Fr(\))32 b Fn(\017)470 469 y Ff(\000)p Fj(n=)p Fm(2)565 490 y Fn(')597 497 y Fj(l)610 490 y Fr(\()p Fn(A)666 497 y Fm(0)685 490 y Fn(;)8 b(B)744 497 y Fm(0)763 490 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))17 b(exp)1076 417 y Fi( )1117 490 y Fk(\000)1170 456 y Fn(\031)r Fr(\()p Fn(c)1240 438 y Fm(2)1240 468 y(2)1259 456 y Fn(y)1285 438 y Fm(2)1283 468 y(2)1315 456 y Fr(+)11 b Fn(d)1389 438 y Fm(2)1389 468 y(3)1409 456 y Fr(\))p 1170 478 259 2 v 1244 524 a(2)p Fn(b)1289 531 y Fm(1)1308 524 y Fn(\021)1334 507 y Fm(0)1332 535 y(1)1441 417 y Fi(!)139 631 y Fk(\002)24 b Fr(exp)294 558 y Fi( )335 631 y Fk(\000)p Fn(i)414 597 y Fk(j)p Fn(d)453 604 y Fm(3)473 597 y Fk(j)p 396 619 111 2 v 396 665 a Fr(2)p Fn(b)441 672 y Fm(1)460 665 y Fn(\021)486 648 y Fm(0)484 676 y(1)519 631 y Fk(h)8 b Fn(y)r(;)16 b(M)e(y)c Fk(i)19 b(\000)h Fn(i)827 597 y(c)848 579 y Fm(2)848 610 y(2)868 597 y Fn(y)894 579 y Fm(2)892 610 y(2)p 816 619 V 816 665 a Fr(2)p Fn(b)861 672 y Fm(1)880 665 y Fn(\021)906 648 y Fm(0)904 676 y(1)947 631 y Fr(ln)1004 558 y Fi( )1051 597 y Fr(2)p Fn(b)1096 604 y Fm(1)1115 597 y Fn(\021)1141 579 y Fm(0)1139 610 y(1)1161 597 y Fn(s)p 1051 619 134 2 v 1081 665 a Fk(j)p Fn(d)1120 672 y Fm(3)1140 665 y Fk(j)1197 558 y Fi(!)8 b(!)139 772 y Fk(\002)24 b Fr(exp)294 699 y Fi( )335 772 y Fn(i)357 738 y(\021)12 b Fk(\001)f Fn(y)p 357 760 88 2 v 390 806 a(\017)468 772 y Fk(\000)19 b Fn(i)548 738 y Fk(j)p Fn(d)587 745 y Fm(3)607 738 y Fk(j)8 b Fn(b)j Fk(\001)g Fn(y)p 548 760 164 2 v 586 806 a(b)607 813 y Fm(1)627 806 y Fn(\021)653 789 y Fm(0)651 817 y(1)735 772 y Fk(\000)20 b Fn(is)8 b(b)i Fk(\001)h Fn(y)932 699 y Fi(!)139 913 y Fk(\002)24 b Fr(exp)294 840 y Fi( )335 913 y Fn(i)360 880 y(S)p 357 902 40 2 v 357 947 a(\017)377 933 y Fm(2)420 913 y Fr(+)19 b Fn(i)499 880 y(S)532 861 y Ff(B)529 892 y Fm(0)558 880 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 499 902 178 2 v 568 947 a Fn(\017)588 933 y Fm(2)701 913 y Fk(\000)19 b Fn(i)781 880 y(b)802 887 y Fm(1)821 880 y Fn(\021)847 861 y Fm(0)845 892 y(1)866 880 y Fn(s)889 861 y Fm(2)p 781 902 129 2 v 832 947 a Fr(2)933 913 y(+)h Fn(i)1045 880 y(d)1070 861 y Fm(2)1070 892 y(3)p 1013 902 111 2 v 1013 947 a Fr(4)p Fn(b)1058 954 y Fm(1)1077 947 y Fn(\021)1103 930 y Fm(0)1101 958 y(1)1147 913 y Fr(+)f Fn(i)1259 880 y(d)1284 861 y Fm(2)1284 892 y(3)p 1226 902 V 1226 947 a Fr(2)p Fn(b)1271 954 y Fm(1)1291 947 y Fn(\021)1317 930 y Fm(0)1315 958 y(1)1358 913 y Fr(ln)1415 840 y Fi( )1474 880 y Fr(2)p Fn(b)1519 887 y Fm(1)1539 880 y Fn(\021)1565 861 y Fm(0)1563 892 y(1)p 1461 902 136 2 v 1461 947 a Fk(j)p Fn(d)1500 954 y Fm(3)1520 947 y Fk(j)p Fn(\017)1554 933 y Fm(2)1574 947 y Fn(s)1610 840 y Fi(!)8 b(!)60 1054 y Fr(=)41 b Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)264 1034 y Fj(\016)281 1022 y Fe(0)293 1054 y Fk(k)p Fn(y)344 1034 y Ff(A)374 1054 y Fk(k)p Fr(\))32 b Fn(\017)470 1034 y Ff(\000)p Fj(n=)p Fm(2)565 1054 y Fn(')597 1061 y Fj(l)610 1054 y Fr(\()p Fn(A)666 1061 y Fm(0)685 1054 y Fn(;)8 b(B)747 1034 y Ff(A)744 1067 y Fm(1)777 1054 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))16 b(exp)1145 981 y Fi( )1186 1054 y Fk(\000)1238 1021 y Fn(\031)r Fr(\()p Fn(c)1308 1003 y Fm(2)1308 1033 y(2)1327 1021 y Fn(y)1353 1003 y Fm(2)1351 1033 y(2)1384 1021 y Fr(+)11 b Fn(d)1458 1003 y Fm(2)1458 1033 y(3)1478 1021 y Fr(\))p 1238 1043 259 2 v 1312 1089 a(2)p Fn(b)1357 1096 y Fm(1)1377 1089 y Fn(\021)1403 1071 y Fm(0)1401 1099 y(1)1510 981 y Fi(!)139 1196 y Fk(\002)24 b Fr(exp)294 1122 y Fi( )335 1196 y Fn(i)357 1162 y(\021)12 b Fk(\001)f Fn(y)p 357 1184 88 2 v 390 1230 a(\017)468 1196 y Fk(\000)19 b Fn(i)548 1162 y Fk(j)p Fn(d)587 1169 y Fm(3)607 1162 y Fk(j)8 b Fn(b)j Fk(\001)g Fn(y)p 548 1184 164 2 v 586 1230 a(b)607 1237 y Fm(1)627 1230 y Fn(\021)653 1213 y Fm(0)651 1241 y(1)735 1196 y Fk(\000)20 b Fn(is)8 b(b)i Fk(\001)h Fn(y)932 1122 y Fi(!)139 1337 y Fk(\002)24 b Fr(exp)294 1264 y Fi( )335 1337 y Fn(i)360 1303 y(S)p 357 1325 40 2 v 357 1371 a(\017)377 1356 y Fm(2)420 1337 y Fr(+)19 b Fn(i)499 1303 y(S)532 1285 y Ff(B)529 1315 y Fm(0)558 1303 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 499 1325 178 2 v 568 1371 a Fn(\017)588 1356 y Fm(2)701 1337 y Fk(\000)19 b Fn(i)781 1303 y(b)802 1310 y Fm(1)821 1303 y Fn(\021)847 1285 y Fm(0)845 1315 y(1)866 1303 y Fn(s)889 1285 y Fm(2)p 781 1325 129 2 v 832 1371 a Fr(2)933 1337 y(+)h Fn(i)1045 1303 y(d)1070 1285 y Fm(2)1070 1315 y(3)p 1013 1325 111 2 v 1013 1371 a Fr(4)p Fn(b)1058 1378 y Fm(1)1077 1371 y Fn(\021)1103 1354 y Fm(0)1101 1382 y(1)1147 1337 y Fr(+)f Fn(i)1259 1303 y(d)1284 1285 y Fm(2)1284 1315 y(3)p 1226 1325 V 1226 1371 a Fr(2)p Fn(b)1271 1378 y Fm(1)1291 1371 y Fn(\021)1317 1354 y Fm(0)1315 1382 y(1)1358 1337 y Fr(ln)1415 1264 y Fi( )1474 1303 y Fr(2)p Fn(b)1519 1310 y Fm(1)1539 1303 y Fn(\021)1565 1285 y Fm(0)1563 1315 y(1)p 1461 1325 136 2 v 1461 1371 a Fk(j)p Fn(d)1500 1378 y Fm(3)1520 1371 y Fk(j)p Fn(\017)1554 1356 y Fm(2)1574 1371 y Fn(s)1610 1264 y Fi(!)8 b(!)1692 1337 y Fn(:)0 1449 y Fr(By)16 b(lemm)o(a)e(3.1)i(of)h([20])f(and)h (prop)q(osition)g(2.3,)f(this)g(appro)o(ximately)e(equals)101 1529 y Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)226 1509 y Fj(\016)243 1497 y Fe(0)255 1529 y Fk(k)p Fn(y)306 1509 y Ff(A)336 1529 y Fk(k)p Fr(\))33 b Fn(\017)433 1509 y Ff(\000)p Fj(n=)p Fm(2)559 1488 y Fi(X)535 1582 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)670 1513 y Fi(e)660 1529 y Fn(d)685 1509 y Ff(A)685 1542 y Fj(m)727 1529 y Fn(')759 1536 y Fj(m)793 1529 y Fr(\()p Fn(A)849 1509 y Ff(A)878 1529 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)996 1509 y Ff(A)1026 1529 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))101 1686 y Fk(\002)24 b Fr(exp)256 1613 y Fi( )297 1686 y Fn(i)319 1652 y(\021)12 b Fk(\001)f Fn(y)p 319 1674 88 2 v 352 1720 a(\017)430 1686 y Fk(\000)19 b Fn(i)510 1652 y Fk(j)p Fn(d)549 1659 y Fm(3)569 1652 y Fk(j)8 b Fn(b)j Fk(\001)g Fn(y)p 510 1674 164 2 v 549 1720 a(b)570 1727 y Fm(1)589 1720 y Fn(\021)615 1703 y Fm(0)613 1731 y(1)698 1686 y Fk(\000)19 b Fn(is)8 b(b)i Fk(\001)h Fn(y)894 1613 y Fi(!)101 1827 y Fk(\002)24 b Fr(exp)256 1754 y Fi( )297 1827 y Fn(i)322 1793 y(S)p 319 1815 40 2 v 319 1861 a(\017)339 1847 y Fm(2)382 1827 y Fr(+)19 b Fn(i)461 1793 y(S)494 1775 y Ff(B)491 1805 y Fm(0)520 1793 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 461 1815 178 2 v 530 1861 a Fn(\017)550 1847 y Fm(2)663 1827 y Fk(\000)19 b Fn(i)743 1793 y(b)764 1800 y Fm(1)783 1793 y Fn(\021)809 1775 y Fm(0)807 1805 y(1)828 1793 y Fn(s)851 1775 y Fm(2)p 743 1815 129 2 v 795 1861 a Fr(2)895 1827 y(+)h Fn(i)1007 1793 y(d)1032 1775 y Fm(2)1032 1805 y(3)p 975 1815 111 2 v 975 1861 a Fr(4)p Fn(b)1020 1868 y Fm(1)1039 1861 y Fn(\021)1065 1844 y Fm(0)1063 1872 y(1)1109 1827 y Fr(+)f Fn(i)1221 1793 y(d)1246 1775 y Fm(2)1246 1805 y(3)p 1188 1815 V 1188 1861 a Fr(2)p Fn(b)1233 1868 y Fm(1)1253 1861 y Fn(\021)1279 1844 y Fm(0)1277 1872 y(1)1320 1827 y Fr(ln)1377 1754 y Fi( )1436 1793 y Fr(2)p Fn(b)1481 1800 y Fm(1)1501 1793 y Fn(\021)1527 1775 y Fm(0)1525 1805 y(1)p 1423 1815 136 2 v 1423 1861 a Fk(j)p Fn(d)1462 1868 y Fm(3)1482 1861 y Fk(j)p Fn(\017)1516 1847 y Fm(2)1536 1861 y Fn(s)1572 1754 y Fi(!)8 b(!)1654 1827 y Fn(;)157 b Fr(\(3.75\))0 1948 y(where)16 b(the)g(n)o(um)o(b)q(ers)430 1934 y(~)422 1948 y Fn(d)447 1930 y Ff(A)447 1960 y Fj(m)497 1948 y Fr(are)g(de\014ned)g(b)o(y)g(the)g(relation)18 2071 y(exp)109 1998 y Fi( )142 2071 y Fk(\000)204 2037 y Fn(\031)r(d)259 2019 y Fm(2)259 2050 y(3)p 186 2060 111 2 v 186 2105 a Fr(2)p Fn(b)231 2112 y Fm(1)251 2105 y Fn(\021)277 2088 y Fm(0)275 2116 y(1)301 1998 y Fi(!)359 2071 y Fk(H)401 2078 y Fj(l)414 2071 y Fr(\()p Fn(A)470 2078 y Fm(0)489 2071 y Fr(;)g Fk(j)p Fn(A)570 2078 y Fm(0)589 2071 y Fk(j)603 2051 y Ff(\000)p Fm(1)650 2071 y Fn(y)r Fr(\))30 b(=)793 1998 y Fi(")822 2037 y Fr(det)16 b Fn(A)943 2019 y Ff(A)973 2037 y Fr(\(0\))p 822 2060 214 2 v 859 2105 a(det)g Fn(A)980 2112 y Fm(0)1041 1998 y Fi(#)1065 2010 y Fm(1)p Fj(=)p Fm(2)1153 2030 y Fi(X)1128 2124 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)1262 2058 y Fr(~)1254 2071 y Fn(d)1279 2051 y Ff(A)1279 2083 y Fj(m)1320 2071 y Fk(H)1362 2078 y Fj(m)1396 2071 y Fr(\()p Fn(A)1452 2051 y Ff(A)1482 2071 y Fr(\(0\);)g Fk(j)p Fn(A)1625 2051 y Ff(A)1655 2071 y Fr(\(0\))p Fk(j)1731 2051 y Ff(\000)p Fm(1)1778 2071 y Fn(y)r Fr(\))p Fn(:)0 2207 y Fr(One)23 b(ma)o(y)e(see)i(that)g(the)g(sum)f(is)g(\014nite)h(and)g(compute)f (the)1213 2194 y(~)1205 2207 y Fn(d)1230 2189 y Ff(A)1230 2219 y Fj(m)1286 2207 y Fr(b)o(y)h(equating)f(co)q(e\016cien)o(ts)g(of) h(the)0 2267 y(monomials)14 b(in)o(v)o(olv)o(ed)g(in)i(this)g (expression.)73 2328 y(W)l(e)g(no)o(w)h(note)f(the)g(follo)o(wing)g (estimates,)e(the)i(last)h(of)f(whic)o(h)g(follo)o(ws)g(from)f(prop)q (osition)i(2.2:)122 2360 y Fi(\020)155 2408 y Fn(\021)181 2388 y Ff(A)211 2408 y Fr(\(0\))j Fk(\000)f Fn(\021)377 2388 y Fm(0)405 2360 y Fi(\021)471 2408 y Fr(=)42 b Fk(\000)p Fr(2)p Fk(j)p Fn(d)653 2415 y Fm(3)673 2408 y Fk(j)p Fn(\017)19 b Fr(+)g Fk(O)q Fr(\()p Fn(\017)863 2388 y Fm(2)882 2408 y Fr(\))p Fn(;)910 b Fr(\(3.76\))149 2500 y Fn(\021)175 2479 y Ff(A)205 2500 y Fr(\(0\))267 2479 y Fm(2)306 2500 y Fk(\000)19 b Fn(\021)390 2479 y Fm(0)410 2471 y(2)471 2500 y Fr(=)42 b Fk(\000)p Fr(4)p Fk(j)p Fn(d)653 2507 y Fm(3)673 2500 y Fk(j)p Fn(\017)19 b Fr(+)g Fk(O)q Fr(\()p Fn(\017)863 2479 y Fm(2)882 2500 y Fr(\))p Fn(;)910 b Fr(\(3.77\))159 2600 y Fn(\021)185 2580 y Ff(A)215 2600 y Fr(\()p Fn(t)p Fr(\))19 b Fk(\000)g Fn(\021)r Fr(\()p Fn(t)p Fr(\))41 b(=)556 2566 y Fn(\017)p Fk(j)p Fn(d)615 2573 y Fm(3)634 2566 y Fk(j)p 556 2589 93 2 v 559 2634 a Fn(b)580 2641 y Fm(1)599 2634 y Fn(\021)625 2617 y Fm(0)623 2645 y(1)661 2600 y Fn(b)19 b Fk(\000)h Fn(t)8 b(b)19 b Fr(+)g Fk(O)q Fr(\()p Fn(\017)963 2580 y Fm(2)982 2600 y Fn(=t)p Fr(\))h(+)11 b Fk(O)q Fr(\()p Fn(t)1190 2580 y Fm(2)1209 2600 y Fr(\))19 b(+)h Fk(O)q Fr(\()p Fn(\017)1385 2580 y Fm(4)1404 2600 y Fn(=t)1446 2580 y Fm(3)1466 2600 y Fr(\))p Fn(:)326 b Fr(\(3.78\))951 2753 y(39)p eop %%Page: 40 40 40 39 bop 0 -22 a Fr(Using)16 b(these,)g(lemm)o(a)e(3.1)i(of)h([20],)f (and)g(corollary)g(2.2,)g(w)o(e)g(see)g(that)h(\(3.75\))g(appro)o (ximately)d(equals)-3 76 y Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)122 55 y Fj(\016)139 44 y Fe(0)152 76 y Fk(k)p Fn(y)203 55 y Ff(A)232 76 y Fk(k)p Fr(\))33 b Fn(\017)329 55 y Ff(\000)p Fj(n=)p Fm(2)456 34 y Fi(X)431 128 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)565 63 y Fr(~)557 76 y Fn(d)582 55 y Ff(A)582 88 y Fj(m)624 76 y Fn(')656 83 y Fj(m)689 76 y Fr(\()p Fn(A)745 55 y Ff(A)775 76 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)893 55 y Ff(A)922 76 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)1164 55 y Ff(A)1194 76 y Fr(\))-3 250 y Fk(\002)25 b Fr(exp)152 165 y Fi(0)152 240 y(@)196 250 y Fn(i)218 217 y(\021)244 199 y Ff(A)285 217 y Fk(\001)11 b Fn(y)336 199 y Ff(A)p 218 239 148 2 v 282 285 a Fn(\017)390 250 y Fk(\000)20 b Fn(i)471 217 y Fk(j)p Fn(d)510 224 y Fm(3)529 217 y Fk(j)8 b Fn(b)j Fk(\001)g Fn(y)p 471 239 164 2 v 509 285 a(b)530 292 y Fm(1)549 285 y Fn(\021)575 267 y Fm(0)573 296 y(1)658 250 y Fk(\000)19 b Fn(is)8 b(b)j Fk(\001)g Fn(y)20 b Fr(+)g Fn(i)945 159 y Fi(\020)978 207 y Fn(\021)r Fr(\()p Fn(t)p Fr(\))10 b Fk(\000)h Fn(\021)1146 189 y Ff(A)1176 207 y Fr(\()p Fn(t)p Fr(\))1232 159 y Fi(\021)1267 207 y Fk(\001)g Fn(y)p 945 239 374 2 v 1121 285 a(\017)1331 165 y Fi(1)1331 240 y(A)-3 404 y Fk(\002)25 b Fr(exp)152 331 y Fi( )193 404 y Fn(i)218 370 y(S)p 215 392 40 2 v 215 438 a(\017)235 424 y Fm(2)278 404 y Fr(+)20 b Fn(i)358 370 y(S)391 352 y Ff(B)388 383 y Fm(0)416 370 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 358 392 178 2 v 426 438 a Fn(\017)446 424 y Fm(2)559 404 y Fk(\000)19 b Fn(i)639 370 y(b)660 377 y Fm(1)679 370 y Fn(\021)705 352 y Fm(0)703 383 y(1)725 370 y Fn(s)748 352 y Fm(2)p 639 392 129 2 v 691 438 a Fr(2)792 404 y(+)g Fn(i)903 370 y(d)928 352 y Fm(2)928 383 y(3)p 871 392 111 2 v 871 438 a Fr(4)p Fn(b)916 445 y Fm(1)935 438 y Fn(\021)961 421 y Fm(0)959 449 y(1)1005 404 y Fr(+)h Fn(i)1117 370 y(d)1142 352 y Fm(2)1142 383 y(3)p 1085 392 V 1085 438 a Fr(2)p Fn(b)1130 445 y Fm(1)1149 438 y Fn(\021)1175 421 y Fm(0)1173 449 y(1)1216 404 y Fr(ln)1274 331 y Fi( )1332 370 y Fr(2)p Fn(b)1377 377 y Fm(1)1397 370 y Fn(\021)1423 352 y Fm(0)1421 383 y(1)p 1320 392 136 2 v 1320 438 a Fk(j)p Fn(d)1359 445 y Fm(3)1379 438 y Fk(j)p Fn(\017)1413 424 y Fm(2)1432 438 y Fn(s)1468 331 y Fi(!)8 b(!)-3 545 y Fk(\002)25 b Fr(exp)152 472 y Fi( )193 545 y Fn(i)215 512 y Fk(j)p Fn(d)254 519 y Fm(3)274 512 y Fk(j)p Fn(t)p 215 534 91 2 v 250 579 a(\017)321 545 y Fr(+)11 b Fn(i)392 512 y Fr(\()p Fn(\021)437 493 y Ff(A)467 512 y Fr(\(0\))g Fk(\000)g Fn(\021)616 493 y Fm(0)636 512 y Fr(\))g Fk(\001)f Fn(\021)716 493 y Ff(A)747 512 y Fr(\(0\))p Fn(t)p 392 534 435 2 v 589 579 a(\017)609 565 y Fm(2)843 545 y Fk(\000)g Fn(i)914 512 y(b)935 519 y Fm(1)954 512 y Fn(\021)980 493 y Ff(A)978 524 y Fm(1)1011 512 y Fr(\(0\))p 914 534 159 2 v 962 579 a(2)p Fn(\017)1006 565 y Fm(2)1078 545 y Fn(t)1096 525 y Fm(2)1126 545 y Fk(\000)h Fn(i)1267 512 y(d)1292 493 y Fm(2)1292 524 y(3)p 1198 534 184 2 v 1198 579 a Fr(2)p Fn(b)1243 586 y Fm(1)1263 579 y Fn(\021)1289 562 y Ff(A)1287 590 y Fm(1)1319 579 y Fr(\(0\))1394 472 y Fi(")1432 512 y Fr(1)p 1432 534 25 2 v 1432 579 a(2)1472 545 y(+)g(ln)1579 472 y Fi( )1616 512 y Fr(2)p Fn(b)1661 519 y Fm(1)1681 512 y Fn(\021)1707 493 y Ff(A)1705 524 y Fm(1)1737 512 y Fr(\(0\))p Fn(t)p 1616 534 201 2 v 1671 579 a Fk(j)p Fn(d)1710 586 y Fm(3)1730 579 y Fk(j)p Fn(\017)1822 472 y Fi(!#!)1921 545 y Fn(:)0 677 y Fr(Using)20 b(\(3.76\),)h(\(3.78\),)h(and)e(the)g(de\014nition)g(of)g Fn(\021)952 659 y Ff(A)982 677 y Fr(\(0\),)h(w)o(e)f(see)g(that)g(this) g(quan)o(tit)o(y)f(appro)o(ximately)0 737 y(equals)205 835 y Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)330 814 y Fj(\016)347 803 y Fe(0)359 835 y Fk(k)p Fn(y)410 814 y Ff(A)440 835 y Fk(k)p Fr(\))533 793 y Fi(X)508 887 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)642 822 y Fr(~)634 835 y Fn(d)659 814 y Ff(A)659 847 y Fj(m)709 835 y Fn(')741 842 y Fj(m)774 835 y Fr(\()p Fn(A)830 814 y Ff(A)860 835 y Fr(\()p Fn(t)p Fr(\))p Fn(;)h(B)978 814 y Ff(A)1007 835 y Fr(\()p 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y(W)l(e)17 b(de\014ne)35 b Fn(d)270 1138 y Ff(A)270 1168 y Fj(m)328 1156 y Fr(=)24 b Fk(\000)8 b Fn(e)460 1133 y Fj(i)477 1110 y Fh(S)497 1099 y Fe(B)496 1122 y Fd(0)520 1110 y(\()p Fh(\017;)p Fe(\000)p Fd(\))p Fe(\000)p Fh(S)634 1099 y Fe(A)633 1122 y Fd(0)661 1110 y(\()p Fh(\017;)p Fd(+\))p 477 1125 255 2 v 588 1150 a Fh(\017)601 1143 y Fd(2)148 b Fr(~)755 1156 y Fn(d)780 1138 y Ff(A)780 1168 y Fj(m)814 1156 y Fr(,)17 b(apply)g(corollary)g(2.3,)h(and)g (recall)e(that)i(w)o(e)f(are)h(studying)0 1216 y(the)f(\014rst)h(comp)q (onen)o(t)e(of)i(the)f(inner)g(solution.)25 b(By)17 b(using)h(lemm)o (as)d(3.1)j(and)g(3.2,)g(w)o(e)e(see)i(that)f(\(3.79\))0 1276 y(matc)o(hes)e(the)h(follo)o(wing)g(appro)o(ximate)e(solution)j (to)f(the)g(full)g(Sc)o(hr\177)-24 b(odinger)16 b(equation.)122 1373 y Fn(F)7 b Fr(\()p Fn(\017)200 1353 y Fj(\016)217 1341 y Fe(0)229 1373 y Fk(k)p Fn(y)280 1353 y Ff(A)310 1373 y Fk(k)p Fr(\))16 b Fn(e)393 1353 y Fj(iS)428 1341 y Fe(A)455 1353 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)528 1341 y Fd(2)595 1332 y Fi(X)571 1426 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)696 1373 y Fn(d)721 1353 y Ff(A)721 1386 y Fj(m)763 1373 y Fn(')795 1380 y Fj(m)828 1373 y Fr(\()p Fn(A)884 1353 y Ff(A)914 1373 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1032 1353 y Ff(A)1061 1373 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1159 1353 y Fm(2)1178 1373 y Fn(;)g(a)1226 1353 y Ff(A)1256 1373 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1360 1353 y Ff(A)1389 1373 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))16 b(\010)1565 1353 y Fm(+)1565 1386 y Ff(A)1595 1373 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))p Fn(:)0 1517 y Fr(This)16 b(is)g(the)g(\014rst)h(term)d(of)j (\(3.73\).)73 1607 y(W)l(e)f(no)o(w)h(concen)o(trate)e(on)i(the)f (second)g(comp)q(onen)o(t)g(of)g(\(3.65\),)h(whic)o(h)e(equals)205 1729 y Fn(F)7 b Fr(\()p Fn(\017)283 1709 y Fj(\016)300 1697 y Fe(0)312 1729 y Fk(k)p Fn(y)r Fk(k)p Fr(\))24 b(exp)522 1668 y Fi(\022)553 1729 y Fn(i)578 1695 y(S)p 575 1717 40 2 v 575 1763 a(\017)595 1749 y Fm(2)630 1729 y Fr(+)11 b Fn(i)701 1695 y(\021)h Fk(\001)f Fn(y)p 701 1717 88 2 v 734 1763 a(\017)793 1668 y Fi(\023)848 1729 y Fn(C)883 1736 y Fm(2)902 1729 y Fr(\()p Fn(y)r Fr(\))205 1870 y Fk(\002)265 1837 y Fr(\()p Fk(\000)p Fr(1)g Fk(\000)g Fn(i)p Fr(\)\()p Fn(c)484 1844 y Fm(2)503 1837 y Fn(y)527 1844 y Fm(2)558 1837 y Fk(\000)g Fn(id)650 1844 y Fm(3)669 1837 y Fr(\))p 265 1859 424 2 v 401 1918 a(2)425 1867 y Fi(q)p 467 1867 87 2 v 51 x Fn(b)488 1925 y Fm(1)507 1918 y Fn(\021)533 1901 y Fm(0)531 1929 y(1)710 1870 y Fn(D)750 1877 y Fj(p)768 1883 y Fh(l)780 1877 y Ff(\000)p Fm(1)835 1785 y Fi(0)835 1859 y(@)877 1837 y Fr(\()p Fk(\000)p Fr(1)g Fk(\000)g Fn(i)p Fr(\))p 877 1859 179 2 v 902 1867 a Fi(q)p 944 1867 87 2 v 51 x Fn(b)965 1925 y Fm(1)984 1918 y Fn(\021)1010 1901 y Fm(0)1008 1929 y(1)1060 1870 y Fr(\()p Fn(b)1100 1877 y Fm(1)1120 1870 y Fn(\021)1146 1850 y Fm(0)1144 1883 y(1)1165 1870 y Fn(s)g Fr(+)g Fn(b)1269 1877 y Fm(1)1289 1870 y Fn(y)1313 1877 y Fm(1)1343 1870 y Fr(+)g Fn(b)1413 1877 y Fm(2)1433 1870 y Fn(y)1457 1877 y Fm(2)1476 1870 y Fr(\))1495 1785 y Fi(1)1495 1859 y(A)1540 1870 y Fn(:)0 2019 y Fr(By)16 b(the)g(explicit)e(form)o(ula)g(for)j Fn(C)621 2026 y Fm(2)641 2019 y Fr(\()p Fn(y)r Fr(\))e(and)i(form)o(ula)e(\(3.70\),)h (this)h(appro)o(ximately)d(equals)205 2147 y Fk(\000)8 b Fn(F)f Fr(\()p Fn(\017)330 2127 y Fj(\016)347 2115 y Fe(0)359 2147 y Fk(k)p Fn(y)r Fk(k)p Fr(\))24 b(exp)569 2087 y Fi(\022)600 2147 y Fn(i)625 2114 y(S)p 622 2136 40 2 v 622 2181 a(\017)642 2167 y Fm(2)677 2147 y Fr(+)11 b Fn(i)748 2114 y(\021)h Fk(\001)f Fn(y)p 748 2136 88 2 v 781 2181 a(\017)840 2087 y Fi(\023)895 2147 y Fn(\017)915 2127 y Ff(\000)p Fj(n=)p Fm(2)1009 2147 y Fn(')1041 2154 y Fj(l)1054 2147 y Fr(\()p Fn(A)1110 2154 y Fm(0)1129 2147 y Fn(;)d(B)1188 2154 y Fm(0)1208 2147 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))17 b(exp)1521 2074 y Fi( )1562 2147 y Fk(\000)1614 2114 y Fn(\031)r Fr(\()p Fn(c)1684 2096 y Fm(2)1684 2126 y(2)1703 2114 y Fn(y)1729 2096 y Fm(2)1727 2126 y(2)1759 2114 y Fr(+)11 b Fn(d)1833 2096 y Fm(2)1833 2126 y(3)1854 2114 y Fr(\))p 1614 2136 259 2 v 1688 2181 a(8)p Fn(b)1733 2188 y Fm(1)1753 2181 y Fn(\021)1779 2164 y Fm(0)1777 2192 y(1)1886 2074 y Fi(!)205 2288 y Fk(\002)24 b Fr(exp)351 2215 y Fi( )384 2288 y Fn(i)406 2255 y(S)439 2237 y Ff(B)436 2267 y Fm(0)465 2255 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 406 2277 178 2 v 474 2323 a Fn(\017)494 2308 y Fm(2)599 2288 y Fk(\000)j Fn(i)691 2255 y(d)716 2237 y Fm(2)716 2267 y(3)p 671 2277 87 2 v 671 2323 a Fn(b)692 2330 y Fm(1)711 2323 y Fn(\021)737 2305 y Fm(0)735 2334 y(1)778 2288 y Fr(ln)16 b Fn(\017)j Fr(+)11 b Fn(i)977 2255 y(d)1002 2237 y Fm(2)1002 2267 y(3)p 945 2277 111 2 v 945 2323 a Fr(4)p Fn(b)990 2330 y Fm(1)1010 2323 y Fn(\021)1036 2305 y Fm(0)1034 2334 y(1)1071 2288 y Fk(\000)g Fn(i)1175 2255 y(d)1200 2237 y Fm(2)1200 2267 y(3)p 1143 2277 V 1143 2323 a Fr(2)p Fn(b)1188 2330 y Fm(1)1208 2323 y Fn(\021)1234 2305 y Fm(0)1232 2334 y(1)1275 2288 y Fr(ln)16 b Fk(j)p Fn(d)1371 2295 y Fm(3)1391 2288 y Fk(j)j Fr(+)11 b Fn(i)1515 2255 y Fr(3)p Fn(d)1564 2237 y Fm(2)1564 2267 y(3)p 1495 2277 V 1495 2323 a Fr(4)p Fn(b)1540 2330 y Fm(1)1559 2323 y Fn(\021)1585 2305 y Fm(0)1583 2334 y(1)1627 2288 y Fr(ln)d(\(2)p Fn(b)1740 2295 y Fm(1)1759 2288 y Fn(\021)1785 2268 y Fm(0)1783 2301 y(1)1805 2288 y Fr(\))1824 2215 y Fi(!)205 2430 y Fk(\002)24 b Fr(exp)351 2357 y Fi( )392 2430 y Fk(\000)p Fn(i)465 2396 y(c)486 2378 y Fm(2)486 2408 y(2)506 2396 y Fn(y)532 2378 y Fm(2)530 2408 y(2)p 453 2418 V 453 2464 a Fr(4)p Fn(b)498 2471 y Fm(1)517 2464 y Fn(\021)543 2447 y Fm(0)541 2475 y(1)585 2430 y Fr(ln)8 b(\(2)p Fn(b)698 2437 y Fm(1)717 2430 y Fn(\021)743 2409 y Fm(0)741 2442 y(1)763 2430 y Fr(\))j(+)g Fn(i)876 2396 y(c)897 2378 y Fm(2)897 2408 y(2)917 2396 y Fn(y)943 2378 y Fm(2)941 2408 y(2)p 864 2418 V 864 2464 a Fr(2)p Fn(b)909 2471 y Fm(1)928 2464 y Fn(\021)954 2447 y Fm(0)952 2475 y(1)996 2430 y Fr(ln)16 b Fk(j)p Fn(d)1092 2437 y Fm(3)1112 2430 y Fk(j)j(\000)11 b Fn(i)1217 2396 y Fk(j)p Fn(d)1256 2403 y Fm(3)1275 2396 y Fk(j)d Fn(b)j Fk(\001)g Fn(y)p 1217 2418 164 2 v 1255 2464 a(b)1276 2471 y Fm(1)1295 2464 y Fn(\021)1321 2447 y Fm(0)1319 2475 y(1)1393 2357 y Fi(!)205 2569 y Fk(\002)265 2536 y Fr(\()p Fk(\000)p Fr(1)g Fk(\000)g Fn(i)p Fr(\)\()p Fn(c)484 2543 y Fm(2)503 2536 y Fn(y)527 2543 y Fm(2)558 2536 y Fk(\000)g Fn(id)650 2543 y Fm(3)669 2536 y Fr(\))p 265 2558 424 2 v 401 2566 a Fi(q)p 442 2566 111 2 v 442 2617 a Fr(2)p Fn(b)487 2624 y Fm(1)507 2617 y Fn(\021)533 2600 y Fm(0)531 2628 y(1)891 2500 y Fk(p)p 932 2500 30 2 v 932 2536 a Fn(\031)p 747 2558 359 2 v 747 2616 a Fr(\000)785 2568 y Fi(\020)811 2616 y Fr(1)g(+)g Fn(i)917 2590 y Fm(\()p Fj(c)946 2579 y Fd(2)946 2601 y(2)963 2590 y Fj(y)981 2579 y Fd(2)980 2601 y(2)999 2590 y Fm(+)p Fj(d)1044 2579 y Fd(2)1044 2601 y(3)1062 2590 y Fm(\))p 917 2605 159 2 v 953 2634 a(2)p Fj(b)986 2639 y Fd(1)1003 2634 y Fj(\021)1022 2623 y Fd(0)1021 2645 y(1)1080 2568 y Fi(\021)951 2753 y Fr(40)p eop %%Page: 41 41 41 40 bop 205 3 a Fk(\002)24 b Fr(exp)359 -70 y Fi( )401 3 y Fn(i)423 -30 y Fr(\()p Fn(b)463 -23 y Fm(1)482 -30 y Fn(\021)508 -48 y Fm(0)506 -18 y(1)527 -30 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))712 -48 y Fm(2)p 423 -8 310 2 v 521 37 a Fr(2)p Fn(b)566 44 y Fm(1)586 37 y Fn(\021)612 20 y Fm(0)610 48 y(1)756 3 y Fk(\000)819 -30 y Fn(\031)r Fr(\()p Fn(c)889 -48 y Fm(2)889 -18 y(2)908 -30 y Fn(y)934 -48 y Fm(2)932 -18 y(2)964 -30 y Fr(+)g Fn(d)1038 -48 y Fm(2)1038 -18 y(3)1058 -30 y Fr(\))p 819 -8 259 2 v 893 37 a(8)p Fn(b)938 44 y Fm(1)958 37 y Fn(\021)984 20 y Fm(0)982 48 y(1)1090 -70 y Fi(!)205 148 y Fk(\002)24 b Fr(exp)359 75 y Fi( )401 148 y Fn(i)423 115 y Fr(\()p Fn(c)463 97 y Fm(2)463 127 y(2)482 115 y Fn(y)508 97 y Fm(2)506 127 y(2)538 115 y Fr(+)11 b Fn(d)612 97 y Fm(2)612 127 y(3)632 115 y Fr(\))p 423 137 230 2 v 481 183 a(2)p Fn(b)526 190 y Fm(1)546 183 y Fn(\021)572 165 y Fm(0)570 193 y(1)673 148 y Fr(ln)730 75 y Fi( )763 73 y(s)p 804 73 96 2 v 840 115 a Fr(2)p 809 137 87 2 v 809 183 a Fn(b)830 190 y Fm(1)850 183 y Fn(\021)876 165 y Fm(0)874 193 y(1)917 148 y Fr(\()p Fn(b)957 155 y Fm(1)976 148 y Fn(\021)1002 128 y Fm(0)1000 161 y(1)1022 148 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)f Fn(y)r Fr(\))1214 75 y Fi(!)e(!)1297 148 y Fn(:)514 b Fr(\(3.80\))0 283 y(As)12 b(with)g(\(3.74\),)h(w)o(e)f(drop)h(the)20 b Fn(b)s Fk(\001)s Fn(y)h Fr(in)12 b(the)g(logarithm)f(in)h(the)g (\014nal)g(factor)g(and)h(w)o(e)f(replace)f Fn(F)c Fr(\()p Fn(\017)1799 265 y Ff(\000)p Fj(\016)1843 253 y Fe(0)1856 283 y Fk(k)p Fn(y)r Fk(k)p Fr(\))0 343 y(b)o(y)16 b Fn(F)7 b Fr(\()p Fn(\017)146 325 y Ff(\000)p Fj(\016)190 313 y Fe(0)202 343 y Fk(k)p Fn(y)253 325 y Ff(B)279 343 y Fk(k)p Fr(\).)21 b(Th)o(us,)16 b(simple)e(calculations)i(sho)o(w)h (that)f(\(3.80\))h(appro)o(ximately)d(equals)205 470 y Fn(F)7 b Fr(\()p Fn(\017)283 449 y Fj(\016)300 437 y Fe(0)312 470 y Fk(k)p Fn(y)363 449 y Ff(B)389 470 y Fk(k)p Fr(\))481 400 y Fi(s)p 523 400 121 2 v 568 436 a Fn(\031)p 528 458 111 2 v 528 504 a Fr(2)p Fn(b)573 511 y Fm(1)593 504 y Fn(\021)619 487 y Fm(0)617 515 y(1)697 436 y Fr(\(1)k(+)g Fn(i)p Fr(\)\()p Fn(c)876 443 y Fm(2)896 436 y Fn(y)920 443 y Fm(2)950 436 y Fk(\000)g Fn(id)1042 443 y Fm(3)1062 436 y Fr(\))p 697 458 384 2 v 710 517 a(\000)748 469 y Fi(\020)773 517 y Fr(1)h(+)f Fn(i)880 491 y Fm(\()p Fj(c)909 479 y Fd(2)909 502 y(2)926 491 y Fj(y)944 479 y Fd(2)943 502 y(2)962 491 y Fm(+)p Fj(d)1007 479 y Fd(2)1007 502 y(3)1024 491 y Fm(\))p 880 505 159 2 v 916 534 a(2)p Fj(b)949 539 y Fd(1)966 534 y Fj(\021)985 523 y Fd(0)984 546 y(1)1043 469 y Fi(\021)205 637 y Fk(\002)16 b Fn(\017)280 616 y Ff(\000)p Fj(n=)p Fm(2)374 637 y Fn(')406 644 y Fj(l)419 637 y Fr(\()p Fn(A)475 644 y Fm(0)494 637 y Fn(;)8 b(B)553 644 y Fm(0)573 637 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))17 b(exp)886 564 y Fi( )927 637 y Fk(\000)979 603 y Fn(\031)r Fr(\()p Fn(c)1049 585 y Fm(2)1049 615 y(2)1068 603 y Fn(y)1094 585 y Fm(2)1092 615 y(2)1125 603 y Fr(+)11 b Fn(d)1199 585 y Fm(2)1199 615 y(3)1219 603 y Fr(\))p 979 625 259 2 v 1053 671 a(4)p Fn(b)1098 678 y Fm(1)1118 671 y Fn(\021)1144 654 y Fm(0)1142 682 y(1)1251 564 y Fi(!)205 778 y Fk(\002)24 b Fr(exp)359 705 y Fi( )401 778 y Fn(i)441 744 y Fk(j)p Fn(d)480 751 y Fm(3)500 744 y Fk(j)p 423 766 111 2 v 423 812 a Fr(2)p Fn(b)468 819 y Fm(1)487 812 y Fn(\021)513 795 y Fm(0)511 823 y(1)546 778 y Fk(h)8 b Fn(y)r(;)16 b(M)e(y)c Fk(i)20 b Fr(+)f Fn(i)854 744 y(c)875 726 y Fm(2)875 757 y(2)894 744 y Fn(y)920 726 y Fm(2)918 757 y(2)p 842 766 V 842 812 a Fr(2)p Fn(b)887 819 y Fm(1)906 812 y Fn(\021)932 795 y Fm(0)930 823 y(1)973 778 y Fr(ln)1031 705 y Fi( )1077 744 y Fr(2)p Fn(b)1122 751 y Fm(1)1142 744 y Fn(\021)1168 726 y Fm(0)1166 757 y(1)1187 744 y Fn(s)p 1077 766 134 2 v 1107 812 a Fk(j)p Fn(d)1146 819 y Fm(3)1166 812 y Fk(j)1224 705 y Fi(!)1276 778 y Fr(+)g Fn(i)1367 744 y(c)1388 726 y Fm(2)1388 757 y(2)1408 744 y Fn(y)1434 726 y Fm(2)1432 757 y(2)p 1355 766 111 2 v 1355 812 a Fr(2)p Fn(b)1400 819 y Fm(1)1420 812 y Fn(\021)1446 795 y Fm(0)1444 823 y(1)1487 778 y Fr(ln)1544 705 y Fi( )1623 744 y Fn(d)1648 726 y Fm(2)1648 757 y(3)p 1590 766 V 1590 812 a Fr(2)p Fn(b)1635 819 y Fm(1)1655 812 y Fn(\021)1681 795 y Fm(0)1679 823 y(1)1714 705 y Fi(!)8 b(!)205 919 y Fk(\002)24 b Fr(exp)359 846 y Fi( )401 919 y Fn(i)423 885 y(\021)12 b Fk(\001)f Fn(y)p 423 907 88 2 v 456 953 a(\017)534 919 y Fk(\000)19 b Fn(i)614 885 y Fk(j)p Fn(d)653 892 y Fm(3)673 885 y Fk(j)8 b Fn(b)j Fk(\001)f Fn(y)p 614 907 164 2 v 652 953 a(b)673 960 y Fm(1)693 953 y Fn(\021)719 936 y Fm(0)717 964 y(1)801 919 y Fr(+)20 b Fn(is)8 b(b)i Fk(\001)h Fn(y)997 846 y Fi(!)205 1067 y Fk(\002)24 b Fr(exp)359 994 y Fi( )401 1067 y Fn(i)426 1033 y(S)p 423 1055 40 2 v 423 1101 a(\017)443 1087 y Fm(2)486 1067 y Fr(+)19 b Fn(i)565 1033 y(S)598 1015 y Ff(B)595 1046 y Fm(0)624 1033 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 565 1055 178 2 v 634 1101 a Fn(\017)654 1087 y Fm(2)766 1067 y Fr(+)20 b Fn(i)846 1033 y(b)867 1040 y Fm(1)886 1033 y Fn(\021)912 1015 y Fm(0)910 1046 y(1)931 1033 y Fn(s)954 1015 y Fm(2)p 846 1055 129 2 v 897 1101 a Fr(2)998 1067 y(+)g Fn(i)1110 1033 y(d)1135 1015 y Fm(2)1135 1046 y(3)p 1078 1055 111 2 v 1078 1101 a Fr(4)p Fn(b)1123 1108 y Fm(1)1142 1101 y Fn(\021)1168 1084 y Fm(0)1166 1112 y(1)1212 1067 y Fr(+)f Fn(i)1324 1033 y(d)1349 1015 y Fm(2)1349 1046 y(3)p 1291 1055 V 1291 1101 a Fr(2)p Fn(b)1336 1108 y Fm(1)1356 1101 y Fn(\021)1382 1084 y Fm(0)1380 1112 y(1)1423 1067 y Fr(ln)1480 994 y Fi( )1518 1033 y Fr(4)p Fn(b)1563 1015 y Fm(2)1563 1046 y(1)1583 1033 y Fn(\021)1609 1015 y Fm(0)1607 1046 y(1)1629 1007 y(2)1648 1033 y Fn(s)p 1518 1055 154 2 v 1539 1101 a Fk(j)p Fn(d)1578 1108 y Fm(3)1597 1101 y Fk(j)p Fn(\017)1631 1087 y Fm(2)1676 994 y Fi(!)8 b(!)1759 1067 y Fn(:)0 1195 y Fr(By)16 b(lemm)o(a)e(3.1)i(of)h([20])f(and)h (prop)q(osition)g(2.3,)f(this)g(appro)o(ximately)e(equals)205 1322 y Fn(F)7 b Fr(\()p Fn(\017)283 1301 y Fj(\016)300 1290 y Fe(0)312 1322 y Fk(k)p Fn(y)363 1301 y Ff(B)389 1322 y Fk(k)p Fr(\))481 1252 y Fi(s)p 523 1252 121 2 v 568 1288 a Fn(\031)p 528 1310 111 2 v 528 1356 a Fr(2)p Fn(b)573 1363 y Fm(1)593 1356 y Fn(\021)619 1339 y Fm(0)617 1367 y(1)697 1288 y Fr(\(1)k(+)g Fn(i)p Fr(\)\()p Fn(c)876 1295 y Fm(2)896 1288 y Fn(y)920 1295 y Fm(2)950 1288 y Fk(\000)g Fn(id)1042 1295 y Fm(3)1062 1288 y Fr(\))p 697 1310 384 2 v 710 1369 a(\000)748 1321 y Fi(\020)773 1369 y Fr(1)h(+)f Fn(i)880 1343 y Fm(\()p Fj(c)909 1331 y Fd(2)909 1354 y(2)926 1343 y Fj(y)944 1331 y Fd(2)943 1354 y(2)962 1343 y Fm(+)p Fj(d)1007 1331 y Fd(2)1007 1354 y(3)1024 1343 y Fm(\))p 880 1357 159 2 v 916 1386 a(2)p Fj(b)949 1391 y Fd(1)966 1386 y Fj(\021)985 1375 y Fd(0)984 1398 y(1)1043 1321 y Fi(\021)205 1489 y Fk(\002)16 b Fn(\017)280 1468 y Ff(\000)p Fj(n=)p Fm(2)374 1489 y Fn(')406 1496 y Fj(l)419 1489 y Fr(\()p Fn(A)475 1468 y Ff(B)501 1489 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)619 1468 y Ff(B)644 1489 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))16 b(exp)1012 1416 y Fi( )1053 1489 y Fk(\000)1105 1455 y Fn(\031)r Fr(\()p Fn(c)1175 1437 y Fm(2)1175 1468 y(2)1194 1455 y Fn(y)1220 1437 y Fm(2)1218 1468 y(2)1251 1455 y Fr(+)11 b Fn(d)1325 1437 y Fm(2)1325 1468 y(3)1345 1455 y Fr(\))p 1105 1477 259 2 v 1179 1523 a(4)p Fn(b)1224 1530 y Fm(1)1244 1523 y Fn(\021)1270 1506 y Fm(0)1268 1534 y(1)1377 1416 y Fi(!)205 1630 y Fk(\002)24 b Fr(exp)359 1557 y Fi( )401 1630 y Fn(i)423 1596 y(\021)12 b Fk(\001)f Fn(y)p 423 1618 88 2 v 456 1664 a(\017)534 1630 y Fk(\000)19 b Fn(i)614 1596 y Fk(j)p Fn(d)653 1603 y Fm(3)673 1596 y Fk(j)8 b Fn(b)j Fk(\001)f Fn(y)p 614 1618 164 2 v 652 1664 a(b)673 1671 y Fm(1)693 1664 y Fn(\021)719 1647 y Fm(0)717 1675 y(1)801 1630 y Fr(+)20 b Fn(is)8 b(b)i Fk(\001)h Fn(y)997 1557 y Fi(!)205 1778 y Fk(\002)24 b Fr(exp)359 1705 y Fi( )401 1778 y Fn(i)426 1744 y(S)p 423 1766 40 2 v 423 1812 a(\017)443 1798 y Fm(2)486 1778 y Fr(+)19 b Fn(i)565 1744 y(S)598 1726 y Ff(B)595 1757 y Fm(0)624 1744 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 565 1766 178 2 v 634 1812 a Fn(\017)654 1798 y Fm(2)766 1778 y Fr(+)20 b Fn(i)846 1744 y(b)867 1751 y Fm(1)886 1744 y Fn(\021)912 1726 y Fm(0)910 1757 y(1)931 1744 y Fn(s)954 1726 y Fm(2)p 846 1766 129 2 v 897 1812 a Fr(2)998 1778 y(+)g Fn(i)1110 1744 y(d)1135 1726 y Fm(2)1135 1757 y(3)p 1078 1766 111 2 v 1078 1812 a Fr(4)p Fn(b)1123 1819 y Fm(1)1142 1812 y Fn(\021)1168 1795 y Fm(0)1166 1823 y(1)1212 1778 y Fr(+)f Fn(i)1324 1744 y(d)1349 1726 y Fm(2)1349 1757 y(3)p 1291 1766 V 1291 1812 a Fr(2)p Fn(b)1336 1819 y Fm(1)1356 1812 y Fn(\021)1382 1795 y Fm(0)1380 1823 y(1)1423 1778 y Fr(ln)1480 1705 y Fi( )1518 1744 y Fr(4)p Fn(b)1563 1726 y Fm(2)1563 1757 y(1)1583 1744 y Fn(\021)1609 1726 y Fm(0)1607 1757 y(1)1629 1718 y(2)1648 1744 y Fn(s)p 1518 1766 154 2 v 1539 1812 a Fk(j)p Fn(d)1578 1819 y Fm(3)1597 1812 y Fk(j)p Fn(\017)1631 1798 y Fm(2)1676 1705 y Fi(!)8 b(!)1759 1778 y Fn(:)0 1906 y Fr(Using)j(lemma)e(3.1)j(of)g([20],)g(lemm)o(as)e(2.1)i(and)g (2.2,)g(and)h(corollary)e(2.2,)h(w)o(e)f(see)h(that)g(this)f(appro)o (ximately)0 1966 y(equals)243 2091 y Fn(F)c Fr(\()p Fn(\017)321 2071 y Fj(\016)338 2059 y Fe(0)350 2091 y Fk(k)p Fn(y)401 2071 y Ff(B)427 2091 y Fk(k)p Fr(\))519 2021 y Fi(s)p 561 2021 121 2 v 606 2058 a Fn(\031)p 566 2080 111 2 v 566 2126 a Fr(2)p Fn(b)611 2133 y Fm(1)631 2126 y Fn(\021)657 2108 y Fm(0)655 2136 y(1)749 2058 y Fr(\(1)k(+)g Fn(i)p Fr(\)\()p Fn(c)928 2065 y Fm(2)948 2058 y Fn(y)974 2040 y Ff(B)972 2070 y Fm(2)1010 2058 y Fk(\000)g Fn(id)1102 2065 y Fm(3)1122 2058 y Fr(\))p 735 2080 421 2 v 735 2147 a(\000)773 2086 y Fi(\022)804 2147 y Fr(1)h(+)f Fn(i)911 2121 y Fm(\()p Fj(c)940 2109 y Fd(2)940 2132 y(2)957 2121 y Fm(\()p Fj(y)989 2109 y Fe(B)988 2132 y Fd(2)1012 2121 y Fm(\))1026 2111 y Fd(2)1043 2121 y Fm(+)p Fj(d)1088 2109 y Fd(2)1088 2132 y(3)1106 2121 y Fm(\))p 911 2135 210 2 v 972 2165 a(2)p Fj(b)1005 2170 y Fd(1)1022 2165 y Fj(\021)1041 2153 y Fd(0)1040 2176 y(1)1124 2086 y Fi(\023)243 2276 y Fk(\002)16 b Fn(\017)318 2255 y Ff(\000)p Fj(n=)p Fm(2)412 2276 y Fn(')444 2283 y Fj(l)457 2276 y Fr(\()p Fn(A)513 2255 y Ff(B)539 2276 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)657 2255 y Ff(B)682 2276 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)924 2255 y Ff(B)949 2276 y Fr(\))17 b(exp)1076 2203 y Fi( )1117 2276 y Fk(\000)1169 2242 y Fn(\031)r Fr(\()p Fn(c)1239 2224 y Fm(2)1239 2254 y(2)1258 2242 y Fr(\()p Fn(y)1303 2224 y Ff(B)1301 2254 y Fm(2)1329 2242 y Fr(\))1348 2224 y Fm(2)1379 2242 y Fr(+)11 b Fn(d)1453 2224 y Fm(2)1453 2254 y(3)1473 2242 y Fr(\))p 1169 2264 323 2 v 1275 2310 a(4)p Fn(b)1320 2317 y Fm(1)1340 2310 y Fn(\021)1366 2293 y Fm(0)1364 2321 y(1)1505 2203 y Fi(!)243 2435 y Fk(\002)24 b Fr(exp)397 2350 y Fi(0)397 2424 y(@)442 2435 y Fn(i)464 2401 y(\021)490 2383 y Ff(B)527 2401 y Fk(\001)10 b Fn(y)577 2383 y Ff(B)p 464 2423 140 2 v 523 2469 a Fn(\017)627 2435 y Fk(\000)20 b Fn(i)708 2401 y Fk(j)p Fn(d)747 2408 y Fm(3)766 2401 y Fk(j)8 b Fn(b)j Fk(\001)g Fn(y)p 708 2423 164 2 v 746 2469 a(b)767 2476 y Fm(1)786 2469 y Fn(\021)812 2452 y Fm(0)810 2480 y(1)895 2435 y Fr(+)19 b Fn(is)8 b(b)j Fk(\001)g Fn(y)21 b Fr(+)e Fn(i)1181 2344 y Fi(\020)1205 2392 y Fn(\021)r Fr(\()p Fn(t)p Fr(\))11 b Fk(\000)g Fn(\021)1374 2374 y Ff(B)1400 2392 y Fr(\()p Fn(t)p Fr(\))1456 2344 y Fi(\021)1491 2392 y Fk(\001)g Fn(y)p 1181 2423 361 2 v 1351 2469 a(\017)1555 2350 y Fi(1)1555 2424 y(A)243 2595 y Fk(\002)24 b Fr(exp)397 2522 y Fi( )438 2595 y Fn(i)463 2562 y(S)p 460 2584 40 2 v 460 2630 a(\017)480 2615 y Fm(2)524 2595 y Fr(+)19 b Fn(i)603 2562 y(S)636 2544 y Ff(B)633 2574 y Fm(0)662 2562 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 603 2584 178 2 v 672 2630 a Fn(\017)692 2615 y Fm(2)804 2595 y Fr(+)20 b Fn(i)884 2562 y(b)905 2569 y Fm(1)924 2562 y Fn(\021)950 2544 y Fm(0)948 2574 y(1)969 2562 y Fn(s)992 2544 y Fm(2)p 884 2584 129 2 v 935 2630 a Fr(2)1036 2595 y(+)g Fn(i)1148 2562 y(d)1173 2544 y Fm(2)1173 2574 y(3)p 1116 2584 111 2 v 1116 2630 a Fr(4)p Fn(b)1161 2637 y Fm(1)1180 2630 y Fn(\021)1206 2612 y Fm(0)1204 2640 y(1)1250 2595 y Fr(+)f Fn(i)1362 2562 y(d)1387 2544 y Fm(2)1387 2574 y(3)p 1329 2584 V 1329 2630 a Fr(2)p Fn(b)1374 2637 y Fm(1)1394 2630 y Fn(\021)1420 2612 y Fm(0)1418 2640 y(1)1461 2595 y Fr(ln)1518 2522 y Fi( )1556 2562 y Fr(4)p Fn(b)1601 2544 y Fm(2)1601 2574 y(1)1621 2562 y Fn(\021)1647 2544 y Fm(0)1645 2574 y(1)1666 2535 y(2)1686 2562 y Fn(s)p 1556 2584 154 2 v 1577 2630 a Fk(j)p Fn(d)1616 2637 y Fm(3)1635 2630 y Fk(j)p Fn(\017)1669 2615 y Fm(2)1714 2522 y Fi(!)8 b(!)951 2753 y Fr(41)p eop %%Page: 42 42 42 41 bop 243 3 a Fk(\002)24 b Fr(exp)397 -70 y Fi( )438 3 y Fk(\000)p Fn(i)499 -30 y Fk(j)p Fn(d)538 -23 y Fm(3)558 -30 y Fk(j)p Fn(t)p 499 -8 91 2 v 534 37 a(\017)614 3 y Fr(+)19 b Fn(i)693 -30 y(b)714 -23 y Fm(1)733 -30 y Fn(\021)759 -48 y Fm(0)757 -18 y(1)p 693 -8 87 2 v 716 37 a Fn(\017)736 23 y Fm(2)783 3 y Fn(t)801 -17 y Fm(2)840 3 y Fr(+)g Fn(i)952 -30 y(d)977 -48 y Fm(2)977 -18 y(3)p 919 -8 111 2 v 919 37 a Fr(2)p Fn(b)964 44 y Fm(1)984 37 y Fn(\021)1010 20 y Fm(0)1008 48 y(1)1043 -70 y Fi(")1080 -30 y Fr(1)p 1080 -8 25 2 v 1080 37 a(2)1129 3 y(+)g(ln)1244 -70 y Fi( )1281 -30 y Fr(2)p Fn(b)1326 -23 y Fm(1)1346 -30 y Fn(\021)1372 -48 y Fm(0)1370 -18 y(1)1392 -30 y Fn(t)p 1281 -8 129 2 v 1299 37 a Fk(j)p Fn(d)1338 44 y Fm(3)1358 37 y Fk(j)p Fn(\017)1414 -70 y Fi(!)9 b(#)f(!)163 144 y Fr(=)42 b Fn(F)7 b Fr(\()p Fn(\017)321 124 y Fj(\016)338 112 y Fe(0)350 144 y Fk(k)p Fn(y)401 124 y Ff(B)427 144 y Fk(k)p Fr(\))519 74 y Fi(s)p 561 74 121 2 v 606 110 a Fn(\031)p 566 133 111 2 v 566 178 a Fr(2)p Fn(b)611 185 y Fm(1)631 178 y Fn(\021)657 161 y Fm(0)655 189 y(1)749 110 y Fr(\(1)k(+)g Fn(i)p Fr(\)\()p Fn(c)928 117 y Fm(2)948 110 y Fn(y)974 92 y Ff(B)972 123 y Fm(2)1010 110 y Fk(\000)g Fn(id)1102 117 y Fm(3)1122 110 y Fr(\))p 735 133 421 2 v 735 200 a(\000)773 139 y Fi(\022)804 200 y Fr(1)h(+)f Fn(i)911 174 y Fm(\()p Fj(c)940 162 y Fd(2)940 185 y(2)957 174 y Fm(\()p Fj(y)989 162 y Fe(B)988 185 y Fd(2)1012 174 y Fm(\))1026 164 y Fd(2)1043 174 y Fm(+)p Fj(d)1088 162 y Fd(2)1088 185 y(3)1106 174 y Fm(\))p 911 188 210 2 v 972 217 a(2)p Fj(b)1005 222 y Fd(1)1022 217 y Fj(\021)1041 206 y Fd(0)1040 229 y(1)1124 139 y Fi(\023)243 329 y Fk(\002)16 b Fn(')330 336 y Fj(l)343 329 y Fr(\()p Fn(A)399 308 y Ff(B)424 329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)542 308 y Ff(B)567 329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)665 308 y Fm(2)684 329 y Fn(;)g(a)732 308 y Ff(B)758 329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)862 308 y Ff(B)887 329 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))16 b(exp)1119 255 y Fi( )1160 329 y Fk(\000)1212 295 y Fn(\031)r Fr(\()p Fn(c)1282 277 y Fm(2)1282 307 y(2)1301 295 y Fr(\()p Fn(y)1346 277 y Ff(B)1344 307 y Fm(2)1372 295 y Fr(\))1391 277 y Fm(2)1421 295 y Fr(+)c Fn(d)1496 277 y Fm(2)1496 307 y(3)1516 295 y Fr(\))p 1212 317 323 2 v 1318 363 a(4)p Fn(b)1363 370 y Fm(1)1383 363 y Fn(\021)1409 346 y Fm(0)1407 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y(3)p 1134 873 V 1134 919 a Fr(2)p Fn(b)1179 926 y Fm(1)1198 919 y Fn(\021)1224 901 y Fm(0)1222 929 y(1)1257 884 y Fr(ln)1314 811 y Fi( )1361 851 y Fr(8)p Fn(b)1406 833 y Fm(3)1406 863 y(1)1425 851 y Fn(\021)1451 833 y Fm(0)1449 863 y(1)1471 824 y(3)p 1361 873 131 2 v 1403 919 a Fn(d)1428 901 y Fm(2)1428 929 y(3)1504 811 y Fi(!)8 b(!)205 1025 y Fk(\002)260 955 y Fi(s)p 301 955 121 2 v 347 992 a Fn(\031)p 306 1014 111 2 v 306 1060 a Fr(2)p Fn(b)351 1067 y Fm(1)371 1060 y Fn(\021)397 1042 y Fm(0)395 1071 y(1)490 992 y Fr(\(1)j(+)g Fn(i)p Fr(\)\()p Fn(c)669 999 y Fm(2)688 992 y Fn(y)714 974 y Ff(B)712 1004 y Fm(2)751 992 y Fk(\000)g Fn(id)843 999 y Fm(3)863 992 y Fr(\))p 476 1014 421 2 v 476 1081 a(\000)514 1021 y Fi(\022)545 1081 y Fr(1)g(+)g Fn(i)651 1055 y Fm(\()p Fj(c)680 1043 y Fd(2)680 1066 y(2)697 1055 y Fm(\()p Fj(y)729 1043 y Fe(B)728 1066 y Fd(2)753 1055 y Fm(\))767 1045 y Fd(2)784 1055 y Fm(+)p Fj(d)829 1043 y Fd(2)829 1066 y(3)846 1055 y Fm(\))p 651 1069 210 2 v 712 1099 a(2)p 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2614 y Fr(\()p Fn(A)364 2593 y Ff(B)390 2614 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(B)508 2593 y Ff(B)533 2614 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)775 2593 y Ff(B)800 2614 y Fr(\))p Fn(;)17 b(G)p Fr(\()p Fn(y)933 2593 y Ff(B)931 2626 y Fm(2)959 2614 y Fr(\))8 b Fn(K)t Fr(\()p Fn(t)p Fr(\))g Fn(')1127 2621 y Fj(l)1140 2614 y Fr(\()p Fn(A)1196 2593 y Ff(B)1221 2614 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(B)1339 2593 y Ff(B)1364 2614 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)1606 2593 y Ff(B)1632 2614 y Fr(\))p Fk(i)951 2753 y Fr(42)p eop %%Page: 43 43 43 42 bop 72 -5 a Fr(=)42 b Fk(\000)8 b(h)g Fn(')258 2 y Fj(m)292 -5 y Fr(\()p Fn(A)348 -26 y Ff(B)373 -5 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(B)491 -26 y Ff(B)516 -5 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)758 -26 y Ff(B)784 -5 y Fr(\))p Fn(;)842 -39 y(i)p 838 -17 25 2 v 838 29 a Fr(2)884 -53 y Fi(\020)917 -5 y Fr(\001)958 4 y Fj(y)976 -5 y Fe(B)14 b 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Fr(0)p Fn(;)g(y)1786 75 y Ff(B)1812 96 y Fr(\))p Fk(i)p Fn(:)1825 174 y Fr(\(3.83\))0 273 y(The)14 b(\014rst)h(inner)f(pro)q(duct)h(on)f(the)g(righ)o(t)g (hand)h(side)f(of)h(\(3.83\))g(is)f(b)q(ounded)h(b)o(y)f Fn(C)1518 280 y Fj(N)1551 273 y Fr(\(1)7 b(+)g Fk(j)p Fn(m)p Fk(j)p Fr(\))1736 255 y Ff(\000)p Fj(N)1797 273 y Fr(,)14 b(where)0 333 y Fn(N)25 b Fr(is)19 b(arbitrary)g(and)h Fn(C)459 340 y Fj(N)512 333 y Fr(can)g(b)q(e)f(c)o(hosen)g(indep)q (enden)o(t)g(of)g Fn(t)p Fr(.)31 b(The)19 b(second)h(inner)e(pro)q (duct)i(on)g(the)0 393 y(righ)o(t)f(hand)g(side)g(of)g(\(3.83\))h(is)f (b)q(ounded)h(b)o(y)f Fn(C)908 400 y Fj(N)949 393 y Fk(j)8 b Fr(ln\()p Fn(t=\017)p Fr(\))p Fk(j)p Fr(\(1)13 b(+)g Fk(j)p Fn(m)p Fk(j)p Fr(\))1323 375 y Ff(\000)p Fj(N)1402 393 y Fr(where)19 b Fn(N)24 b Fr(is)19 b(arbitrary)g(and)0 454 y Fn(C)35 461 y Fj(N)85 454 y Fr(can)d(b)q(e)h(c)o(hosen)f(indep)q (enden)o(t)f(of)i Fn(t)p Fr(.)73 514 y(By)i(in)o(tegrating)g(these)f (estimates)g(with)h(resp)q(ect)g(to)g 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(appro)o(ximate)e(solution)i(to)g(the)f(full)g(Sc)o(hr\177)-24 b(odinger)12 b(equation.)122 964 y Fn(F)7 b Fr(\()p Fn(\017)200 943 y Fj(\016)217 932 y Fe(0)229 964 y Fk(k)p Fn(y)280 943 y Ff(B)306 964 y Fk(k)p Fr(\))16 b Fn(e)389 943 y Fj(iS)424 932 y Fe(B)447 943 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)520 932 y Fd(2)563 922 y Fi(X)577 1010 y Fj(m)639 964 y Fn(d)664 943 y Ff(B)664 976 y Fj(m)706 964 y Fn(')738 971 y Fj(m)771 964 y Fr(\()p Fn(A)827 943 y Ff(B)853 964 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)971 943 y Ff(B)996 964 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1094 943 y Fm(2)1113 964 y Fn(;)g(a)1161 943 y Ff(B)1186 964 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1290 943 y Ff(B)1315 964 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))16 b(\010)1491 943 y Fm(+)1491 976 y Ff(B)1521 964 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))p Fn(:)0 1094 y Fr(This)16 b(is)g(the)g(second)h(term)d(of)j (\(3.73\).)1201 b Fa(2)0 1184 y Fg(Remarks.)0 1244 y Fr(1.)97 b(Supp)q(ose)17 b(one)g(c)o(ho)q(oses)f Fn(l)f 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Ff(B)1576 1429 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))382 1533 y(+)25 b Fk(O)q Fr(\()p Fn(\017)525 1512 y Fj(p)545 1533 y Fr(\))p Fn(;)0 1635 y Fl(wher)n(e)18 b(the)g Fn(d)244 1617 y Ff(A)244 1647 y Fj(m)295 1635 y Fl(and)f Fn(d)414 1617 y Ff(B)414 1647 y Fj(m)465 1635 y Fl(ar)n(e)g(sp)n(e)n(ci\014e)n(d)g(in)h(the)g(pr)n(o)n(of)e(of)h (lemma)h(3.6.)0 1855 y Fo(4)83 b(T)n(yp)r(e)27 b(4,)g(5,)h(and)f(6)h (Av)n(oided)f(Crossings)0 1965 y Fr(In)c(this)f(section)h(w)o(e)f(mak)o (e)f(precise)h(statemen)o(ts)f(of)j(our)f(results)g(for)g(T)o(yp)q(e)f (4,)j(5,)f(and)g(6)f(Av)o(oided)0 2025 y(Crossings.)34 b(The)20 b(pro)q(ofs)i(of)e(these)g(results)f(are)i(obtained)f(b)o(y)f (simply)f(mimic)n(ki)o(ng)g(argumen)o(ts)h(from)0 2085 y(Section)d(3,)g(so)h(w)o(e)f(do)g(not)h(presen)o(t)f(the)g(details.)0 2229 y Fc(4.1)70 b(T)n(yp)r(e)22 b(4)i(Av)n(oided)e(Crossings)0 2322 y Fr(Since)c(the)i(matrix)d(\(1.15\))j(is)f(the)h(direct)e(sum)g 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Fn(>)h(\017)131 25 y Fm(1)p Ff(\000)p Fj(\030)195 43 y Fl(.)22 b(De\014ne)c Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\))13 b(=)h(max)7 b Fk(f)h(j)p Fn(t)p Fk(j)p Fn(;)16 b(\017)805 25 y Fm(1)p Ff(\000)p Fj(\016)867 14 y Fe(0)880 43 y Fn(=)p Fk(j)p Fn(t)p Fk(j)8 b(g)p Fl(,)17 b(and)g(let)h Fn(a)1203 25 y Ff(C)1225 43 y Fr(\()p Fn(t)p Fr(\))f Fl(and)g Fn(\021)1418 25 y Ff(C)1440 43 y Fr(\()p Fn(t)p Fr(\))g Fl(solve)h(e)n(quations)f(\(2.2\))0 104 y(and)f(\(2.3\).)21 b(F)l(or)15 b Fn(x)g Fl(in)g(the)h(supp)n(ort)f (of)g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)f Fk(\000)g Fn(a)916 85 y Ff(C)937 104 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)1062 85 y Fm(1)p Ff(\000)p Fj(\016)1124 74 y Fe(0)1137 104 y Fr(\))p Fl(,)16 b(ther)n(e)f(exist)h(orthonormal)f(eigenve)n(ctors)0 164 y Fr(\010)35 143 y Ff(\006)35 176 y(C)r Fj(;j)84 164 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))16 b Fl(for)h Fk(C)g Fr(=)c Fk(A)p Fn(;)j Fk(B)r Fl(,)h Fn(t)580 142 y Fj(>)580 181 y(<)612 164 y Fr(0)p Fl(,)h(and)f Fn(j)g Fr(=)d(1)p Fn(;)j 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Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)122 977 y Fr(\010)157 956 y Fm(+)157 989 y Ff(A)p Fj(;)p Fm(2)215 977 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 984 y Fm(3)534 977 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 1049 y Fr(\010)161 1029 y Fm(+)161 1062 y Ff(B)q Fj(;)p Fm(1)215 1049 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 1056 y Fm(2)534 1049 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 1122 y Fr(\010)161 1101 y Fm(+)161 1134 y Ff(B)q Fj(;)p Fm(2)215 1122 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 1129 y Fm(4)534 1122 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(:)73 1264 y Fr(Our)16 b(main)f(result)h(for)h(T)o(yp)q(e)f(4)g(Av)o (oided)f(Crossings)j(is)e(the)g(follo)o(wing:)0 1375 y Fg(Theorem)h(4.1)24 b Fl(L)n(et)f Fn(h)p Fr(\()p Fn(x;)8 b(\017)p Fr(\))23 b Fl(b)n(e)h(a)g(hamiltonian)h(such)f(that)g Fn(h)1226 1383 y Ff(k)1246 1375 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))23 b Fl(is)g(char)n(acterize)n(d)h(by)g(\(1.14\),)0 1436 y(\(1.15\),)18 b(and)g(\(1.33\).)24 b(L)n(et)18 b Fn(a)538 1418 y Ff(C)560 1436 y Fr(\()p Fn(t)p Fr(\))p Fl(,)g Fn(\021)675 1418 y Ff(C)698 1436 y Fl(,)g Fn(A)768 1418 y Ff(C)790 1436 y Fl(,)g Fn(B)863 1418 y Ff(C)886 1436 y Fl(,)g(and)g Fn(S)1047 1418 y Ff(C)1088 1436 y Fl(b)n(e)g(de\014ne)n(d)i(as)e(in)g(Se)n(ction)h(2.)25 b(F)l(or)18 b Fn(j)g Fr(=)d(1)p Fn(;)i Fr(2)p Fl(,)0 1496 y(let)h Fn( )100 1503 y Fj(j)118 1496 y Fr(\()p Fn(x;)8 b(t)p Fr(\))17 b Fl(b)n(e)h(a)f(solution)h(of)g(the)g(Schr\177) -25 b(odinger)18 b(e)n(quation)g(\(1.29\))f(such)h(that)122 1603 y Fn( )154 1610 y Fj(j)172 1603 y Fr(\()p Fn(x;)8 b Fk(\000)p Fn(T)f Fr(\))40 b(=)h Fl(e)477 1582 y Fj(iS)512 1571 y Fe(B)535 1582 y Fm(\()p Ff(\000)p Fj(T)5 b Fm(\))p Fj(=\017)648 1571 y Fd(2)675 1603 y 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1903 y Fd(2)578 1936 y Fn(')610 1943 y Fj(l)623 1936 y Fr(\()p Fn(A)679 1915 y Ff(B)705 1936 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)823 1915 y Ff(B)848 1936 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)946 1915 y Fm(2)965 1936 y Fn(;)g(a)1013 1915 y Ff(B)1038 1936 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1142 1915 y Ff(B)1167 1936 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))17 b(\010)1344 1915 y Ff(\000)1344 1948 y(B)q Fj(;j)1396 1936 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))27 b(+)i Fk(O)q Fr(\()p Fn(\017)1718 1915 y Fj(p)1737 1936 y Fr(\))p Fn(;)0 2035 y Fl(for)17 b Fk(\000)p Fn(T)j Fk(\024)13 b Fn(t)h Fk(\024)f(\000)p Fn(\017)361 2017 y Fm(1)p Ff(\000)p Fj(\030)425 2035 y Fl(.)22 b(F)l(or)17 b Fk(\000)p Fn(\017)612 2017 y Fm(1)p Ff(\000)p Fj(\030)689 2035 y Fk(\024)d Fn(t)f Fk(\024)h Fn(\017)846 2017 y Fm(1)p Ff(\000)p Fj(\030)910 2035 y Fl(,)j(in)h(the)g(b)n(asis)f Fk(f)8 b Fn( )1265 2042 y Fm(1)1284 2035 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)16 b Fk(\001)8 b(\001)g(\001)g Fn(;)16 b( )1550 2042 y 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Fn(y)1262 2244 y Fm(1)1293 2237 y Fr(+)g Fn(b)1363 2244 y Fm(2)1382 2237 y Fn(y)1406 2244 y Fm(2)1426 2237 y Fr(\))1445 2176 y Fi(\023)613 2343 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 2348 y Fd(2)771 2343 y Fj(y)788 2348 y Fd(2)806 2343 y Ff(\000)p Fj(id)863 2348 y Fd(3)881 2343 y Fm(\))p 613 2354 282 2 v 690 2393 a(2)708 2360 y Fk(p)p 749 2360 69 2 v 749 2393 a Fj(b)764 2398 y Fd(1)782 2393 y Fj(\021)801 2382 y Fd(0)800 2404 y(1)900 2366 y Fn(D)940 2373 y Fj(p)958 2379 y Fh(l)970 2373 y Ff(\000)p Fm(1)1026 2305 y Fi(\022)1061 2343 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1061 2354 112 2 v 1062 2360 a Fk(p)p 1104 2360 69 2 v 33 x Fj(b)1119 2398 y Fd(1)1136 2393 y Fj(\021)1155 2382 y Fd(0)1154 2404 y(1)1178 2366 y Fr(\()p Fn(b)1218 2373 y Fm(1)1238 2366 y Fn(\021)1264 2348 y Fm(0)1262 2378 y(1)1283 2366 y Fn(s)g Fr(+)g Fn(b)1387 2373 y Fm(1)1407 2366 y Fn(y)1431 2373 y Fm(1)1461 2366 y Fr(+)g Fn(b)1531 2373 y Fm(2)1551 2366 y Fn(y)1575 2373 y Fm(2)1594 2366 y Fr(\))1613 2305 y Fi(\023)1114 2473 y Fr(0)1114 2554 y(0)1652 2189 y Fi(1)1652 2262 y(C)1652 2287 y(C)1652 2311 y(C)1652 2336 y(C)1652 2361 y(C)1652 2386 y(C)1652 2411 y(C)1652 2436 y(C)1652 2461 y(C)1652 2487 y(A)399 2629 y Fr(+)18 b Fk(O)q Fr(\()p Fn(\017)535 2608 y Fj(p)554 2629 y Fr(\))12 b Fn(;)951 2753 y Fr(45)p eop %%Page: 46 46 46 45 bop 0 -22 a Fl(and)122 68 y Fn( )154 75 y Fm(2)173 68 y Fr(\()p Fn(x;)8 b(t)p Fr(\))41 b(=)g Fl(e)422 47 y Fj(iS)r Fm(\()p Fj(t)p Fm(\))p Fj(=\017)530 36 y Fd(2)547 47 y Fm(+)p Fj(i\021)q Fm(\()p Fj(t)p Fm(\)\()p Fj(x)p Ff(\000)p Fj(a)p Fm(\()p Fj(t)p Fm(\)\))p Fj(=\017)813 36 y Fd(2)838 68 y Fn(C)873 75 y Fm(2)893 68 y Fr(\()p Fn(y)r Fr(\))499 308 y Fk(\002)564 110 y Fi(0)564 184 y(B)564 208 y(B)564 233 y(B)564 258 y(B)564 283 y(B)564 308 y(B)564 333 y(B)564 358 y(B)564 383 y(B)564 409 y(@)1114 148 y Fr(0)1114 229 y(0)777 316 y Fn(D)817 323 y Fj(p)835 329 y Fh(l)857 255 y Fi(\022)893 293 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 893 304 112 2 v 894 310 a Fk(p)p 935 310 69 2 v 935 343 a Fj(b)950 348 y Fd(1)968 343 y Fj(\021)987 332 y Fd(0)986 354 y(1)1010 316 y Fr(\()p Fn(b)1050 323 y Fm(1)1069 316 y Fn(\021)1095 298 y Fm(0)1093 328 y(1)1115 316 y Fn(s)11 b Fr(+)g Fn(b)1219 323 y Fm(1)1238 316 y Fn(y)1262 323 y Fm(1)1293 316 y Fr(+)g Fn(b)1363 323 y Fm(2)1382 316 y Fn(y)1406 323 y Fm(2)1426 316 y Fr(\))1445 255 y Fi(\023)613 422 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 427 y Fd(2)771 422 y Fj(y)788 427 y Fd(2)806 422 y Fm(+)p Fj(id)863 427 y Fd(3)881 422 y Fm(\))p 613 433 282 2 v 690 472 a(2)708 439 y Fk(p)p 749 439 69 2 v 749 472 a Fj(b)764 477 y Fd(1)782 472 y Fj(\021)801 461 y Fd(0)800 483 y(1)900 445 y Fn(D)940 452 y Fj(p)958 458 y Fh(l)970 452 y Ff(\000)p Fm(1)1026 384 y Fi(\022)1061 422 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1061 433 112 2 v 1062 439 a Fk(p)p 1104 439 69 2 v 33 x Fj(b)1119 477 y Fd(1)1136 472 y Fj(\021)1155 461 y Fd(0)1154 483 y(1)1178 445 y Fr(\()p 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Fn(d)744 918 y Ff(B)744 951 y Fj(m;j)812 939 y Fn(')844 946 y Fj(m)877 939 y Fr(\()p Fn(A)933 918 y Ff(B)958 939 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1076 918 y Ff(B)1101 939 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1199 918 y Fm(2)1218 939 y Fn(;)g(a)1266 918 y Ff(B)1292 939 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1396 918 y Ff(B)1421 939 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1589 918 y Fm(+)1589 951 y Ff(B)q Fj(;j)1641 939 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 1043 y(+)26 b Fk(O)q Fr(\()p Fn(\017)542 1022 y Fj(p)561 1043 y Fr(\))p Fn(;)0 1132 y Fl(wher)n(e)18 b(the)g Fn(d)244 1114 y Ff(A)244 1144 y Fj(m;j)321 1132 y Fl(and)f Fn(d)440 1114 y Ff(B)440 1144 y Fj(m;j)517 1132 y Fl(ar)n(e)g(c)n(ompute)n(d)g(by)h(the)g(pr)n (o)n(c)n(ess)e(use)n(d)h(in)h(the)g(pr)n(o)n(of)e(of)h(lemma)h(3.6.)0 1312 y Fc(4.2)70 b(T)n(yp)r(e)22 b(5)i(Av)n(oided)e(Crossings)0 1405 y Fr(T)o(yp)q(e)17 b(5)h(Av)o(oided)e(Crossings)i(ha)o(v)o(e)f (more)f(complicated)f(electronic)g(structure,)i(so)h(w)o(e)f(presen)o (t)g(a)g(few)0 1465 y(more)f(of)i(the)f(details.)23 b(T)l(o)18 b(sp)q(ecify)f(the)g(static)g(eigen)o(v)o(ectors)e(analogous)20 b(to)d(those)h(in)f(lemm)o(a)e(3.1,)i(w)o(e)0 1525 y(de\014ne)h(a)i(co) q(ordinate)f(system)e(b)o(y)h(the)h(follo)o(wing)f(relations,)h(with)g (0)f Fk(\024)g Fn(r)q Fr(,)i(0)e Fk(\024)g Fn(\022)h(<)f(\031)r Fr(,)h(0)f Fk(\024)g Fn(\036)g(<)1910 1505 y Fj(\031)p 1910 1513 22 2 v 1912 1542 a Fm(2)1936 1525 y Fr(,)0 1585 y(and)f(0)d Fk(\024)g Fn(\026)g(<)g Fr(2)p Fn(\031)r Fr(:)236 1684 y Fn(r)43 b Fr(=)380 1632 y Fi(q)p 422 1632 795 2 v 52 x Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))561 1670 y Fm(2)590 1684 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))775 1670 y Fm(2)805 1684 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))986 1670 y Fm(2)1016 1684 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1198 1670 y Fm(2)1217 1684 y Fn(;)122 1757 y(\014)s Fr(\()p Fn(x;)g(\017)p Fr(\))40 b(=)h Fn(r)18 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y(e)511 2247 y Ff(\000)p Fj(i\026)590 2265 y Fr(cos)677 2246 y Fj(\022)p 677 2254 V 677 2283 a Fm(2)716 2265 y Fr(sin)f Fn(\036)642 2326 y Fr(0)602 2386 y(sin)684 2366 y Fj(\022)p 684 2374 V 684 2403 a Fm(2)829 2173 y Fi(1)829 2246 y(C)829 2271 y(C)829 2295 y(C)829 2322 y(A)874 2295 y Fn(;)122 2542 y Fr(\010)157 2522 y Ff(\000)157 2555 y(A)p Fj(;)p Fm(2)215 2542 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 2420 y Fi(0)443 2493 y(B)443 2518 y(B)443 2542 y(B)443 2569 y(@)488 2452 y Fk(\000)p Fn(e)550 2434 y Fj(i\026)601 2452 y Fr(cos)688 2433 y Fj(\022)p 688 2441 V 688 2469 a Fm(2)727 2452 y Fr(sin)17 b Fn(\036)542 2512 y Fr(cos)629 2493 y Fj(\022)p 628 2501 V 628 2530 a Fm(2)668 2512 y Fr(cos)g Fn(\036)584 2573 y Fk(\000)8 b Fr(sin)713 2553 y Fj(\022)p 713 2561 V 713 2590 a Fm(2)648 2633 y Fr(0)841 2420 y Fi(1)841 2493 y(C)841 2518 y(C)841 2542 y(C)841 2569 y(A)885 2542 y Fn(;)951 2753 y Fr(46)p eop %%Page: 47 47 47 46 bop 126 56 a Fr(\010)161 36 y Ff(\000)161 69 y(B)q Fj(;)p Fm(1)215 56 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 -67 y Fi(0)443 6 y(B)443 31 y(B)443 56 y(B)443 83 y(@)579 -34 y Fk(\000)8 b Fr(sin)707 -54 y Fj(\022)p 707 -46 18 2 v 707 -17 a Fm(2)642 26 y Fr(0)488 86 y Fn(e)511 68 y Ff(\000)p Fj(i\026)590 86 y Fr(cos)677 67 y Fj(\022)p 677 75 V 677 104 a Fm(2)716 86 y Fr(sin)16 b Fn(\036)536 147 y Fr(cos)623 127 y Fj(\022)p 623 135 V 623 164 a Fm(2)662 147 y Fr(cos)h Fn(\036)829 -67 y Fi(1)829 6 y(C)829 31 y(C)829 56 y(C)829 83 y(A)874 56 y Fn(;)126 303 y Fr(\010)161 283 y Ff(\000)161 316 y(B)q Fj(;)p Fm(2)215 303 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 180 y Fi(0)443 253 y(B)443 278 y(B)443 303 y(B)443 330 y(@)628 213 y Fr(0)565 273 y Fk(\000)8 b Fr(sin)694 254 y Fj(\022)p 694 262 V 694 290 a Fm(2)499 333 y Fk(\000)g Fr(cos)633 314 y Fj(\022)p 633 322 V 633 351 a Fm(2)672 333 y Fr(cos)17 b Fn(\036)488 394 y(e)511 375 y Fj(i\026)562 394 y Fr(cos)649 374 y Fj(\022)p 649 382 V 649 411 a Fm(2)688 394 y Fr(sin)g Fn(\036)802 180 y Fi(1)802 253 y(C)802 278 y(C)802 303 y(C)802 330 y(A)847 303 y Fn(;)0 489 y Fr(for)g Fn(\031)r(=)p Fr(2)d Fn(<)f(\022)i Fk(\024)f Fn(\031)r Fr(,)h(and)122 675 y(\010)157 654 y Fm(+)157 687 y Ff(A)p Fj(;)p Fm(1)215 675 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 552 y Fi(0)443 625 y(B)443 650 y(B)443 675 y(B)443 701 y(@)597 584 y Fr(cos)684 565 y Fj(\022)p 684 573 V 684 602 a Fm(2)639 645 y Fr(0)488 705 y Fn(e)511 687 y Ff(\000)p Fj(i\026)590 705 y Fr(sin)671 685 y Fj(\022)p 671 693 V 671 722 a Fm(2)710 705 y Fr(sin)17 b Fn(\036)536 765 y Fr(sin)618 745 y Fj(\022)p 617 753 V 617 782 a Fm(2)657 765 y Fr(cos)g Fn(\036)824 552 y Fi(1)824 625 y(C)824 650 y(C)824 675 y(C)824 701 y(A)869 675 y Fn(;)122 922 y Fr(\010)157 901 y Fm(+)157 934 y Ff(A)p Fj(;)p Fm(2)215 922 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 799 y Fi(0)443 872 y(B)443 897 y(B)443 922 y(B)443 948 y(@)626 831 y Fr(0)583 892 y(cos)670 872 y Fj(\022)p 670 880 V 670 909 a Fm(2)499 952 y Fk(\000)8 b Fr(sin)627 932 y Fj(\022)p 627 940 V 627 969 a Fm(2)666 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1325 y Fk(\000)8 b Fr(sin)660 1306 y Fj(\022)p 660 1314 V 660 1343 a Fm(2)700 1325 y Fr(cos)16 b Fn(\036)488 1386 y Fk(\000)p Fn(e)550 1368 y Ff(\000)p Fj(i\026)628 1386 y Fr(sin)710 1366 y Fj(\022)p 710 1374 V 710 1403 a Fm(2)749 1386 y Fr(sin)h Fn(\036)659 1446 y Fr(0)616 1506 y(cos)703 1486 y Fj(\022)p 703 1494 V 703 1523 a Fm(2)863 1293 y Fi(1)863 1366 y(C)863 1391 y(C)863 1416 y(C)863 1442 y(A)907 1416 y Fn(;)0 1601 y Fr(for)g(0)d Fk(\024)f Fn(\022)i(<)f(\031)r(=)p Fr(2.)0 1662 y(Using)e(these)f(de\014nitions)h(and)g(mimic)n(ki)o(ng)e (the)h(pro)q(ofs)j(of)e(lemm)o(as)e(3.1)i(and)h(3.2,)f(w)o(e)g(ha)o(v)o (e)f(the)g(follo)o(wing)0 1722 y(analog)17 b(of)g(lemma)c(4.1:)0 1836 y Fg(Lemma)j(4.2)24 b Fl(Consider)18 b(a)h(system)f(with)h(a)g(T)l (yp)n(e)f(5)g(A)o(voide)n(d)h(Cr)n(ossing.)25 b(Supp)n(ose)19 b Fr(0)d Fn(<)g(\016)1733 1818 y Ff(0)1760 1836 y Fn(<)g(\030)j(<)1912 1816 y Fm(1)p 1912 1824 V 1912 1853 a(3)1935 1836 y Fl(,)0 1901 y Fk(j)p Fn(t)p Fk(j)13 b Fn(>)h(\017)131 1883 y Fm(1)p Ff(\000)p Fj(\030)195 1901 y Fl(.)22 b(De\014ne)c Fn(\026)p Fr(\()p Fn(\017;)8 b(t)p Fr(\))13 b(=)h(max)7 b Fk(f)h(j)p Fn(t)p Fk(j)p Fn(;)16 b(\017)805 1883 y Fm(1)p Ff(\000)p Fj(\016)867 1871 y Fe(0)880 1901 y Fn(=)p Fk(j)p Fn(t)p Fk(j)8 b(g)p Fl(,)17 b(and)g(let)h Fn(a)1203 1883 y Ff(C)1225 1901 y Fr(\()p Fn(t)p Fr(\))f Fl(and)g Fn(\021)1418 1883 y Ff(C)1440 1901 y Fr(\()p Fn(t)p Fr(\))g Fl(solve)h(e)n(quations)f(\(2.2\))0 1961 y(and)f(\(2.3\).)21 b(F)l(or)15 b Fn(x)g Fl(in)g(the)h(supp)n(ort)f(of)g Fn(F)7 b Fr(\()p Fk(k)p Fn(x)f Fk(\000)g Fn(a)916 1943 y Ff(C)937 1961 y Fr(\()p Fn(t)p Fr(\))p Fk(k)p Fn(=\017)1062 1943 y Fm(1)p Ff(\000)p Fj(\016)1124 1931 y Fe(0)1137 1961 y Fr(\))p Fl(,)16 b(ther)n(e)f(exist)h(orthonormal)f(eigenve)n (ctors)0 2021 y Fr(\010)35 2001 y Ff(\006)35 2034 y(C)r Fj(;j)84 2021 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))16 b Fl(for)h Fk(C)g Fr(=)d Fk(A)p Fn(;)i Fk(B)r Fl(,)h Fn(t)581 1999 y Fj(>)581 2039 y(<)613 2021 y Fr(0)p Fl(,)h(and)g Fn(j)f Fr(=)d(1)p Fn(;)j Fr(2)p Fl(,)h(that)g(c)n(orr)n(esp)n(ond)e(to) h Fn(E)1403 2028 y Ff(C)1426 2021 y Fr(\()p Fn(x;)f(\017)p Fr(\))h Fl(and)h(solve)h(\(4.1\).)j(If)0 2086 y(in)c(addition,)f Fk(j)p Fn(t)p Fk(j)c Fn(<)h(\017)396 2068 y Fj(\024)436 2086 y Fl(for)j(some)g Fn(\024)d(>)737 2066 y Fm(1)p 737 2074 V 737 2103 a(2)759 2086 y Fl(,)k(then)g(for)f Fn(t)c(<)h Fr(0)p Fl(,)122 2188 y Fr(\010)157 2167 y Ff(\000)157 2200 y(A)p Fj(;)p Fm(1)215 2188 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 2195 y Fm(4)534 2188 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)122 2260 y Fr(\010)157 2240 y Ff(\000)157 2272 y(A)p Fj(;)p Fm(2)215 2260 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2267 y Fm(3)573 2260 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 2333 y Fr(\010)161 2312 y Ff(\000)161 2345 y(B)q Fj(;)p Fm(1)215 2333 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2340 y Fm(1)573 2333 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 2405 y Fr(\010)161 2385 y Ff(\000)161 2418 y(B)q Fj(;)p Fm(2)215 2405 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2412 y Fm(2)573 2405 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)0 2507 y Fl(and)18 b(for)e Fn(t)e(>)g Fr(0)p Fl(,)122 2609 y Fr(\010)157 2588 y Fm(+)157 2621 y Ff(A)p Fj(;)p Fm(1)215 2609 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 2616 y Fm(1)534 2609 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)951 2753 y Fr(47)p eop %%Page: 48 48 48 47 bop 122 -22 a Fr(\010)157 -42 y Fm(+)157 -9 y Ff(A)p Fj(;)p Fm(2)215 -22 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 -15 y Fm(2)534 -22 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 51 y Fr(\010)161 30 y Fm(+)161 63 y Ff(B)q Fj(;)p Fm(1)215 51 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 58 y Fm(3)534 51 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 124 y Fr(\010)161 103 y Fm(+)161 136 y Ff(B)q Fj(;)p Fm(2)215 124 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fn( )515 131 y Fm(4)534 124 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(:)73 268 y Fr(Our)16 b(main)f(result)h(for)h(T)o(yp)q(e)f(5)g(Av)o(oided)f (Crossings)j(is)e(the)g(follo)o(wing:)0 382 y Fg(Theorem)h(4.2)24 b Fl(L)n(et)f Fn(h)p Fr(\()p Fn(x;)8 b(\017)p Fr(\))23 b Fl(b)n(e)h(a)g(hamiltonian)h(such)f(that)g Fn(h)1226 390 y Ff(k)1246 382 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))23 b Fl(is)g(char)n(acterize)n(d)h(by)g(\(1.14\),)0 442 y(\(1.19\),)17 b(and)h(\(1.37\).)23 b(L)n(et)17 b Fn(a)535 424 y Ff(C)557 442 y Fr(\()p Fn(t)p Fr(\))p Fl(,)h Fn(\021)672 424 y Ff(C)694 442 y Fl(,)g Fn(A)764 424 y Ff(C)786 442 y Fl(,)g Fn(B)859 424 y Ff(C)881 442 y Fl(,)f(and)h Fn(S)1041 424 y Ff(C)1082 442 y Fl(b)n(e)g(the)g(quantities)h(fr)n(om)e(Se)n (ction)h(2.7)g(appr)n(o-)0 502 y(priate)13 b(for)f(T)l(yp)n(e)h(5)g(A)o (voide)n(d)g(Cr)n(ossings.)20 b(F)l(or)13 b Fn(j)k Fr(=)c(1)p Fn(;)k Fr(2)p Fl(,)d(let)g Fn( )1158 509 y Fj(j)1176 502 y Fr(\()p Fn(x;)8 b(t)p Fr(\))k Fl(b)n(e)i(a)f(solution)h(of)f(the) g(Schr\177)-25 b(odinger)0 563 y(e)n(quation)18 b(\(1.29\))f(such)h (that)122 670 y Fn( )154 677 y Fj(j)172 670 y Fr(\()p Fn(x;)8 b Fk(\000)p Fn(T)f Fr(\))40 b(=)h Fl(e)477 649 y Fj(iS)512 638 y Fe(B)535 649 y Fm(\()p Ff(\000)p Fj(T)5 b Fm(\))p Fj(=\017)648 638 y Fd(2)675 670 y Fn(')707 677 y Fj(l)720 670 y Fr(\()p Fn(A)776 649 y Ff(B)802 670 y Fr(\()p Fk(\000)p Fn(T)i Fr(\))p Fn(;)h(B)977 649 y Ff(B)1001 670 y Fr(\()p Fk(\000)p Fn(T)f Fr(\))p Fn(;)h(\017)1156 649 y Fm(2)1174 670 y Fn(;)g(a)1222 649 y Ff(B)1248 670 y Fr(\()p Fk(\000)p Fn(T)f Fr(\))p Fn(;)h(\021)1409 649 y Ff(B)1434 670 y Fr(\()p Fk(\000)p Fn(T)f Fr(\))p Fn(;)h(x)p Fr(\))16 b(\010)1667 649 y Ff(\000)1667 682 y(B)q Fj(;j)1719 670 y Fr(\()p Fn(x;)8 b Fk(\000)p Fn(T)t(;)g(\017)p Fr(\))454 743 y(+)26 b Fk(O)q Fr(\()p Fn(\017)598 722 y Fj(q)617 743 y Fr(\))0 844 y Fl(for)19 b(some)h(p)n(ositive)g Fn(q)r Fl(.)30 b(Then,)21 b(for)e(any)h Fr(0)f Fn(<)f(\030)k(<)c Fr(1)p Fn(=)p Fr(3)p Fl(,)k(ther)n(e)e(exists)g(a)g(p)n(ositive)g Fn(p)h Fl(such)f(that)g(in)g(the)0 905 y(limit)e Fn(\017)13 b Fk(!)h Fr(0)p Fl(,)k(we)g(have)122 1006 y Fn( )154 1013 y Fj(j)172 1006 y Fr(\()p Fn(x;)8 b(t)p Fr(\))41 b(=)g Fl(e)420 986 y Fj(iS)455 974 y Fe(B)479 986 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)552 974 y Fd(2)578 1006 y Fn(')610 1013 y Fj(l)623 1006 y Fr(\()p Fn(A)679 986 y Ff(B)705 1006 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)823 986 y Ff(B)848 1006 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)946 986 y Fm(2)965 1006 y Fn(;)g(a)1013 986 y Ff(B)1038 1006 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1142 986 y Ff(B)1167 1006 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))17 b(\010)1344 986 y Ff(\000)1344 1019 y(B)q Fj(;j)1396 1006 y Fr(\()p Fn(x;)8 b(t;)g(\017)p Fr(\))27 b(+)i Fk(O)q Fr(\()p Fn(\017)1718 986 y Fj(p)1737 1006 y Fr(\))p Fn(;)0 1108 y Fl(for)17 b Fk(\000)p Fn(T)j Fk(\024)13 b Fn(t)h Fk(\024)f(\000)p Fn(\017)361 1090 y Fm(1)p Ff(\000)p Fj(\030)425 1108 y Fl(.)22 b(F)l(or)17 b Fk(\000)p Fn(\017)612 1090 y Fm(1)p Ff(\000)p Fj(\030)689 1108 y Fk(\024)d Fn(t)f Fk(\024)h Fn(\017)846 1090 y Fm(1)p Ff(\000)p Fj(\030)910 1108 y Fl(,)j(in)h(the)g(b)n(asis)f Fk(f)8 b Fn( )1265 1115 y Fm(1)1284 1108 y Fr(\()p Fn(x;)g(\017)p Fr(\))p Fn(;)16 b Fk(\001)8 b(\001)g(\001)g Fn(;)16 b( )1550 1115 y Fm(4)1570 1108 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))g Fk(g)p Fl(,)17 b(we)h(have)122 1214 y Fn( )154 1221 y Fm(1)173 1214 y Fr(\()p Fn(x;)8 b(t)p Fr(\))41 b(=)g Fl(e)422 1194 y Fj(iS)r Fm(\()p Fj(t)p Fm(\))p Fj(=\017)530 1182 y Fd(2)547 1194 y Fm(+)p Fj(i\021)q Fm(\()p Fj(t)p Fm(\)\()p Fj(x)p Ff(\000)p Fj(a)p Fm(\()p Fj(t)p Fm(\)\))p Fj(=\017)813 1182 y Fd(2)848 1214 y Fn(C)883 1221 y Fm(2)902 1214 y Fr(\()p Fn(y)r Fr(\))499 1486 y Fk(\002)564 1263 y Fi(0)564 1336 y(B)564 1361 y(B)564 1386 y(B)564 1411 y(B)564 1436 y(B)564 1461 y(B)564 1486 y(B)564 1510 y(B)564 1535 y(B)564 1560 y(B)564 1585 y(B)564 1612 y(@)794 1311 y Fn(D)834 1318 y Fj(p)852 1324 y Fh(l)875 1251 y Fi(\022)910 1288 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 910 1300 112 2 v 911 1305 a Fk(p)p 953 1305 69 2 v 33 x Fj(b)968 1343 y Fd(1)985 1338 y Fj(\021)1004 1327 y Fd(0)1003 1350 y(1)1027 1311 y Fr(\()p Fn(b)1067 1318 y Fm(1)1087 1311 y Fn(\021)1113 1293 y Fm(0)1111 1324 y(1)1132 1311 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1317 1251 y Fi(\023)1058 1410 y Fr(0)613 1474 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 1479 y Fd(2)771 1474 y Fj(y)788 1479 y Fd(2)806 1474 y Ff(\000)p Fj(id)863 1479 y Fd(3)881 1474 y Fj(y)898 1479 y Fd(3)916 1474 y Fm(\))p 613 1486 317 2 v 707 1524 a(2)725 1491 y Fk(p)p 767 1491 69 2 v 33 x Fj(b)782 1529 y Fd(1)799 1524 y Fj(\021)818 1513 y Fd(0)817 1536 y(1)935 1497 y Fn(D)975 1504 y Fj(p)993 1510 y Fh(l)1005 1504 y Ff(\000)p Fm(1)1060 1437 y Fi(\022)1096 1474 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1096 1486 112 2 v 1097 1491 a Fk(p)p 1138 1491 69 2 v 1138 1524 a Fj(b)1153 1529 y Fd(1)1171 1524 y Fj(\021)1190 1513 y Fd(0)1189 1536 y(1)1213 1497 y Fr(\()p Fn(b)1253 1504 y Fm(1)1272 1497 y Fn(\021)1298 1479 y Fm(0)1296 1510 y(1)1318 1497 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1503 1437 y Fi(\023)699 1632 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p Fj(e)827 1637 y Fd(4)p 699 1644 146 2 v 707 1682 a Fm(2)725 1649 y Fk(p)p 767 1649 69 2 v 33 x Fj(b)782 1687 y Fd(1)799 1682 y Fj(\021)818 1671 y Fd(0)817 1694 y(1)849 1655 y Fn(D)889 1662 y Fj(p)907 1668 y Fh(l)919 1662 y Ff(\000)p Fm(1)975 1595 y Fi(\022)1010 1632 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1010 1644 112 2 v 1011 1649 a Fk(p)p 1053 1649 69 2 v 33 x Fj(b)1068 1687 y Fd(1)1085 1682 y Fj(\021)1104 1671 y Fd(0)1103 1694 y(1)1127 1655 y Fr(\()p Fn(b)1167 1662 y Fm(1)1187 1655 y Fn(\021)1213 1637 y Fm(0)1211 1668 y(1)1232 1655 y Fn(s)h Fr(+)f Fn(b)f Fk(\001)h Fn(y)r Fr(\))1417 1595 y Fi(\023)1541 1263 y(1)1541 1336 y(C)1541 1361 y(C)1541 1386 y(C)1541 1411 y(C)1541 1436 y(C)1541 1461 y(C)1541 1486 y(C)1541 1510 y(C)1541 1535 y(C)1541 1560 y(C)1541 1585 y(C)1541 1612 y(A)399 1753 y Fr(+)18 b Fk(O)q Fr(\()p Fn(\017)535 1732 y Fj(p)554 1753 y Fr(\))12 b Fn(;)0 1855 y Fl(and)122 1956 y Fn( )154 1963 y Fm(2)173 1956 y Fr(\()p Fn(x;)c(t)p Fr(\))41 b(=)g Fl(e)422 1936 y Fj(iS)r Fm(\()p Fj(t)p Fm(\))p Fj(=\017)530 1924 y Fd(2)547 1936 y Fm(+)p Fj(i\021)q Fm(\()p Fj(t)p Fm(\)\()p Fj(x)p Ff(\000)p Fj(a)p Fm(\()p Fj(t)p Fm(\)\))p Fj(=\017)813 1924 y Fd(2)838 1956 y Fn(C)873 1963 y Fm(2)893 1956 y Fr(\()p Fn(y)r Fr(\))499 2229 y Fk(\002)564 2007 y Fi(0)564 2080 y(B)564 2105 y(B)564 2130 y(B)564 2154 y(B)564 2179 y(B)564 2204 y(B)564 2229 y(B)564 2254 y(B)564 2279 y(B)564 2304 y(B)564 2329 y(B)564 2355 y(@)1131 2035 y Fr(0)867 2122 y Fn(D)907 2129 y Fj(p)925 2135 y Fh(l)947 2061 y Fi(\022)983 2099 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 983 2110 112 2 v 984 2116 a Fk(p)p 1025 2116 69 2 v 1025 2149 a Fj(b)1040 2154 y Fd(1)1058 2149 y Fj(\021)1077 2137 y Fd(0)1076 2160 y(1)1100 2122 y Fr(\()p Fn(b)1140 2129 y Fm(1)1159 2122 y Fn(\021)1185 2104 y Fm(0)1183 2134 y(1)1205 2122 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1390 2061 y Fi(\023)785 2228 y Fm(\(1+)p Fj(i)p Fm(\))p Fj(e)886 2233 y Fd(4)p 780 2239 128 2 v 780 2278 a Fm(2)798 2245 y Fk(p)p 839 2245 69 2 v 839 2278 a Fj(b)854 2283 y Fd(1)872 2278 y Fj(\021)891 2266 y Fd(0)890 2289 y(1)913 2251 y Fn(D)953 2258 y Fj(p)971 2264 y Fh(l)983 2258 y Ff(\000)p Fm(1)1039 2190 y Fi(\022)1074 2228 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1074 2239 112 2 v 1075 2245 a Fk(p)p 1117 2245 69 2 v 33 x Fj(b)1132 2283 y Fd(1)1149 2278 y Fj(\021)1168 2266 y Fd(0)1167 2289 y(1)1191 2251 y Fr(\()p Fn(b)1231 2258 y Fm(1)1251 2251 y Fn(\021)1277 2233 y Fm(0)1275 2263 y(1)1296 2251 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1481 2190 y Fi(\023)613 2357 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 2362 y Fd(2)771 2357 y Fj(y)788 2362 y Fd(2)806 2357 y Fm(+)p Fj(id)863 2362 y Fd(3)881 2357 y Fj(y)898 2362 y Fd(3)916 2357 y Fm(\))p 613 2368 317 2 v 707 2407 a(2)725 2374 y Fk(p)p 767 2374 69 2 v 33 x Fj(b)782 2412 y Fd(1)799 2407 y Fj(\021)818 2395 y Fd(0)817 2418 y(1)935 2380 y Fn(D)975 2387 y Fj(p)993 2393 y Fh(l)1005 2387 y Ff(\000)p Fm(1)1060 2319 y Fi(\022)1096 2357 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1096 2368 112 2 v 1097 2374 a Fk(p)p 1138 2374 69 2 v 1138 2407 a Fj(b)1153 2412 y Fd(1)1171 2407 y Fj(\021)1190 2395 y Fd(0)1189 2418 y(1)1213 2380 y Fr(\()p Fn(b)1253 2387 y Fm(1)1272 2380 y Fn(\021)1298 2362 y Fm(0)1296 2392 y(1)1318 2380 y Fn(s)g Fr(+)g Fn(b)1422 2387 y Fm(1)1441 2380 y Fn(y)1465 2387 y Fm(1)1496 2380 y Fr(+)g Fn(b)1566 2387 y Fm(2)1585 2380 y Fn(y)1609 2387 y Fm(2)1629 2380 y Fr(\))1648 2319 y Fi(\023)1687 2007 y(1)1687 2080 y(C)1687 2105 y(C)1687 2130 y(C)1687 2154 y(C)1687 2179 y(C)1687 2204 y(C)1687 2229 y(C)1687 2254 y(C)1687 2279 y(C)1687 2304 y(C)1687 2329 y(C)1687 2355 y(A)399 2498 y Fr(+)18 b Fk(O)q Fr(\()p Fn(\017)535 2478 y Fj(p)554 2498 y Fr(\))12 b Fn(;)951 2753 y Fr(48)p eop %%Page: 49 49 49 48 bop 0 -22 a Fl(wher)n(e)18 b Fn(C)173 -15 y Fm(2)192 -22 y Fr(\()p Fn(y)r Fr(\))f Fl(is)g(de\014ne)n(d)i(by)18 111 y Fn(C)53 118 y Fm(2)73 111 y Fr(\()p Fn(y)r Fr(\))41 b(=)g Fk(\000)8 b Fn(\017)324 91 y Ff(\000)p Fj(n=)p Fm(2)418 111 y Fn(')450 118 y Fj(l)463 111 y Fr(\()p Fn(A)519 118 y Fm(0)539 111 y Fn(;)g(B)598 118 y Fm(0)617 111 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))17 b(exp)930 38 y Fi( )968 78 y Fk(\000)p Fn(\031)r Fr(\()p Fn(c)1077 59 y Fm(2)1077 90 y(2)1096 78 y Fn(y)1122 59 y Fm(2)1120 90 y(2)1152 78 y Fr(+)11 b Fn(d)1226 59 y Fm(2)1226 90 y(3)1246 78 y Fn(y)1272 59 y Fm(2)1270 90 y(3)1303 78 y Fr(+)g Fn(e)1375 59 y Fm(2)1375 90 y(4)1394 78 y Fr(\))p 968 100 445 2 v 1135 145 a(8)p Fn(b)1180 152 y Fm(1)1200 145 y Fn(\021)1226 128 y Fm(0)1224 156 y(1)1426 38 y Fi(!)257 252 y Fk(\002)26 b Fr(exp)405 179 y Fi( )437 252 y Fn(i)459 219 y(S)492 201 y Ff(B)489 231 y Fm(0)518 219 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 459 241 178 2 v 528 287 a Fn(\017)548 272 y Fm(2)652 252 y Fk(\000)j Fn(i)746 219 y(e)769 201 y Fm(2)769 231 y(4)p 724 241 87 2 v 724 287 a Fn(b)745 294 y Fm(1)764 287 y Fn(\021)790 269 y Fm(0)788 297 y(1)831 252 y Fr(ln)17 b Fn(\017)i Fr(+)11 b Fn(i)1032 219 y(e)1055 201 y Fm(2)1055 231 y(4)p 999 241 111 2 v 999 287 a Fr(4)p Fn(b)1044 294 y Fm(1)1063 287 y Fn(\021)1089 269 y Fm(0)1087 297 y(1)1125 252 y Fk(\000)f Fn(i)1230 219 y(e)1253 201 y Fm(2)1253 231 y(4)p 1196 241 V 1196 287 a Fr(2)p Fn(b)1241 294 y Fm(1)1261 287 y Fn(\021)1287 269 y Fm(0)1285 297 y(1)1328 252 y Fr(ln)16 b Fk(j)p Fn(e)1422 259 y Fm(4)1442 252 y Fk(j)j Fr(+)11 b Fn(i)1567 219 y Fr(3)p Fn(e)1614 201 y Fm(2)1614 231 y(4)p 1546 241 V 1546 287 a Fr(4)p Fn(b)1591 294 y Fm(1)1610 287 y Fn(\021)1636 269 y Fm(0)1634 297 y(1)1677 252 y Fr(ln)d(\(2)p Fn(b)1790 259 y Fm(1)1810 252 y Fn(\021)1836 232 y Fm(0)1834 265 y(1)1856 252 y Fr(\))1875 179 y Fi(!)257 394 y Fk(\002)26 b Fr(exp)405 320 y Fi( )446 394 y Fk(\000)p Fn(i)507 360 y(c)528 342 y Fm(2)528 372 y(2)547 360 y Fn(y)573 342 y Fm(2)571 372 y(2)603 360 y Fr(+)11 b Fn(d)677 342 y Fm(2)677 372 y(3)697 360 y Fn(y)723 342 y Fm(2)721 372 y(3)p 507 382 237 2 v 569 428 a Fr(4)p Fn(b)614 435 y Fm(1)634 428 y Fn(\021)660 411 y Fm(0)658 439 y(1)764 394 y Fr(ln)d(\(2)p Fn(b)877 401 y Fm(1)897 394 y Fn(\021)923 373 y Fm(0)921 406 y(1)942 394 y Fr(\))j(+)g Fn(i)1043 360 y(c)1064 342 y Fm(2)1064 372 y(2)1084 360 y Fn(y)1110 342 y Fm(2)1108 372 y(2)1140 360 y Fr(+)g Fn(d)1214 342 y Fm(2)1214 372 y(3)1234 360 y Fn(y)1260 342 y Fm(2)1258 372 y(3)p 1043 382 V 1106 428 a Fr(2)p Fn(b)1151 435 y Fm(1)1171 428 y Fn(\021)1197 411 y Fm(0)1195 439 y(1)1301 394 y Fr(ln)16 b Fk(j)p Fn(e)1395 401 y Fm(4)1414 394 y Fk(j)k(\000)11 b Fn(i)1520 360 y Fk(j)p Fn(e)1557 367 y Fm(4)1575 360 y Fk(j)p Fn(b)g Fk(\001)g Fn(y)p 1520 382 153 2 v 1552 428 a(b)1573 435 y Fm(1)1593 428 y Fn(\021)1619 411 y Fm(0)1617 439 y(1)1685 320 y Fi(!)1726 394 y Fn(;)0 543 y(p)24 550 y Fj(l)60 543 y Fr(=)22 b Fk(\000)8 b Fn(i)189 517 y Fj(c)204 505 y Fd(2)204 528 y(2)221 517 y Fj(y)239 505 y Fd(2)238 528 y(2)257 517 y Fm(+)p Fj(d)302 505 y Fd(2)302 528 y(3)320 517 y Fj(y)338 505 y Fd(2)337 528 y(3)356 517 y Fm(+)p Fj(e)399 505 y Fd(2)399 528 y(4)p 188 531 229 2 v 259 560 a Fm(2)p Fj(b)292 565 y Fd(1)309 560 y Fj(\021)328 549 y Fd(0)327 571 y(1)422 543 y Fl(,)17 b Fn(y)f Fr(=)d(\()p Fn(x)e Fk(\000)g Fn(a)p Fr(\()p Fn(t)p Fr(\)\))p Fn(=\017)p Fl(,)16 b(and)i Fn(s)c Fr(=)g Fn(t=\017)p Fl(.)22 b(Final)r(ly,)c(for)f Fn(\017)1392 525 y Fm(1)p Ff(\000)p Fj(\030)1470 543 y Fk(\024)c Fn(t)h Fk(\024)f Fn(T)7 b Fl(,)122 666 y Fn( )154 673 y Fj(j)172 666 y Fr(\()p Fn(x;)h(t)p Fr(\))41 b(=)g Fl(e)420 646 y Fj(iS)455 634 y Fe(A)482 646 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)555 634 y Fd(2)615 625 y Fi(X)591 719 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)716 666 y Fn(d)741 646 y Ff(A)741 679 y Fj(m;j)809 666 y Fn(')841 673 y Fj(m)874 666 y Fr(\()p Fn(A)930 646 y Ff(A)960 666 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1078 646 y Ff(A)1107 666 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1205 646 y Fm(2)1224 666 y Fn(;)g(a)1272 646 y Ff(A)1302 666 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1406 646 y Ff(A)1435 666 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1603 646 y Fm(+)1603 679 y Ff(A)p Fj(;j)1659 666 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 797 y(+)17 b Fl(e)476 777 y Fj(iS)511 765 y Fe(B)534 777 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)607 765 y Fd(2)642 756 y Fi(X)656 843 y Fj(m)719 797 y Fn(d)744 774 y Ff(B)q Fj(;)p Fm(1)744 809 y Fj(m;j)812 797 y Fn(')844 804 y Fj(m)877 797 y Fr(\()p Fn(A)933 777 y Ff(B)958 797 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1076 777 y Ff(B)1101 797 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1199 777 y Fm(2)1218 797 y Fn(;)g(a)1266 777 y Ff(B)1292 797 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1396 777 y Ff(B)1421 797 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1589 777 y Fm(+)1589 810 y Ff(B)q Fj(;)p Fm(1)1642 797 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 913 y(+)17 b Fl(e)476 893 y Fj(iS)511 881 y Fe(B)534 893 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)607 881 y Fd(2)642 872 y Fi(X)656 959 y Fj(m)719 913 y Fn(d)744 889 y Ff(B)q Fj(;)p Fm(2)744 925 y Fj(m;j)812 913 y Fn(')844 920 y Fj(m)877 913 y Fr(\()p Fn(A)933 893 y Ff(B)958 913 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1076 893 y Ff(B)1101 913 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1199 893 y Fm(2)1218 913 y Fn(;)g(a)1266 893 y Ff(B)1292 913 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1396 893 y Ff(B)1421 913 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1589 893 y Fm(+)1589 925 y Ff(B)q Fj(;)p Fm(2)1642 913 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 1017 y(+)26 b Fk(O)q Fr(\()p Fn(\017)542 996 y Fj(p)561 1017 y Fr(\))p Fn(;)0 1119 y Fl(wher)n(e)21 b(the)f Fn(d)249 1101 y Ff(A)249 1131 y Fj(m;j)329 1119 y Fl(and)h Fn(d)452 1095 y Ff(B)q Fj(;k)452 1130 y(m;j)532 1119 y Fl(ar)n(e)f(c)n(ompute)n(d)g(by)g(the)h(pr)n(o)n(c)n(ess)e(use)n (d)i(in)f(the)h(pr)n(o)n(of)e(of)h(lemma)h(3.6.)31 b(The)0 1179 y(initial)14 b(c)n(onditions)h(for)e(the)h(semiclassic)n(al)h (quantities)g(in)f(the)g(expr)n(ession)g(for)f Fn( )1492 1186 y Fj(j)1510 1179 y Fr(\()p Fn(x;)8 b(t)p Fr(\))13 b Fl(for)g Fn(t)h Fk(\025)f Fn(\017)1807 1161 y Fm(1)p Ff(\000)p Fj(\030)1885 1179 y Fl(ar)n(e)0 1239 y(the)20 b(analo)n(gs)f(of)g(formulas)g(\(3.71\))g(and)g(\(3.72\))f(after)i(one) f(has)g(taken)i(into)e(ac)n(c)n(ount)h(the)f(appr)n(opriate)0 1299 y(alter)n(ations)f(fr)n(om)e(se)n(ction)i(2.7.)0 1498 y Fc(4.3)70 b(T)n(yp)r(e)22 b(6)i(Av)n(oided)e(Crossings)0 1590 y Fr(T)o(yp)q(e)17 b(6)g(Av)o(oided)f(Crossings)i(ha)o(v)o(e)e(ev) o(en)g(more)g(complicated)e(electronic)i(structure.)23 b(T)l(o)17 b(sp)q(ecify)f(the)0 1650 y(static)j(eigen)o(v)o(ectors)e (analogous)j(to)f(those)g(in)g(lemm)o(a)d(3.1,)k(w)o(e)e(de\014ne)g(a)h (co)q(ordinate)h(system)d(b)o(y)h(the)0 1710 y(follo)o(wing)e (relations,)g(with)g(0)e Fk(\024)f Fn(r)q Fr(,)k(0)d Fk(\024)f Fn(\022)i(<)f(\031)r Fr(,)h(0)g Fk(\024)e Fn(\036)h(<)1103 1691 y Fj(\031)p 1103 1699 22 2 v 1105 1727 a Fm(2)1129 1710 y Fr(,)i(0)e Fk(\024)g Fn(\026)g(<)g Fr(2)p Fn(\031)r Fr(,)h(and)i(0)d Fk(\024)g Fn(!)i(<)e Fr(2)p Fn(\031)r Fr(:)236 1829 y Fn(r)43 b Fr(=)380 1776 y Fi(q)p 422 1776 1006 2 v 53 x Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))561 1815 y Fm(2)590 1829 y Fr(+)j Fn(\015)s Fr(\()p Fn(x;)d(\017)p Fr(\))775 1815 y Fm(2)805 1829 y Fr(+)j Fn(\016)r Fr(\()p Fn(x;)d(\017)p Fr(\))986 1815 y Fm(2)1016 1829 y Fr(+)j Fn(\020)t Fr(\()p Fn(x;)d(\017)p Fr(\))1198 1815 y Fm(2)1228 1829 y Fr(+)j Fn(\030)r Fr(\()p Fn(x;)d(\017)p Fr(\))1408 1815 y Fm(2)1427 1829 y Fn(;)122 1902 y(\014)s Fr(\()p Fn(x;)g(\017)p Fr(\))40 b(=)h Fn(r)18 b Fr(cos)f Fn(\022)q(;)124 1974 y(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(r)18 b Fr(sin)f Fn(\022)h Fr(sin)e Fn(\036)g Fr(cos)h Fn(\026;)129 2047 y(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))40 b(=)h Fn(r)18 b Fr(sin)f Fn(\022)h Fr(sin)e Fn(\036)g Fr(sin)h Fn(\026;)127 2120 y(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(r)18 b Fr(sin)f Fn(\022)h Fr(cos)e Fn(\036)h Fr(cos)g Fn(!)r(;)129 2192 y(\030)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)g Fn(r)18 b Fr(sin)f Fn(\022)h Fr(cos)e Fn(\036)h Fr(sin)f Fn(!)r(;)0 2294 y Fr(where)f Fn(\014)s Fr(\()p Fn(x;)8 b(\017)p Fr(\),)13 b Fn(\015)s Fr(\()p Fn(x;)8 b(\017)p Fr(\),)14 b Fn(\016)r Fr(\()p Fn(x;)8 b(\017)p Fr(\),)14 b Fn(\020)t Fr(\()p Fn(x;)8 b(\017)p Fr(\),)14 b(and)h Fn(\030)r Fr(\()p Fn(x;)8 b(\017)p Fr(\))15 b(are)h(giv)o(en)e(b)o(y)g(\(1.40\).) 22 b(The)15 b(static)g(eigen)o(v)o(ectors)0 2354 y(are)h(then)g(giv)o (en)g(b)o(y)122 2537 y(\010)157 2516 y Ff(\000)157 2549 y(A)p Fj(;)p Fm(1)215 2537 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 2414 y Fi(0)443 2487 y(B)443 2512 y(B)443 2537 y(B)443 2564 y(@)498 2447 y Fn(e)521 2429 y Fj(i!)574 2447 y Fr(cos)661 2427 y Fj(\022)p 661 2435 18 2 v 661 2464 a Fm(2)700 2447 y Fr(cos)17 b Fn(\036)488 2507 y(e)511 2489 y Ff(\000)p Fj(i\026)590 2507 y Fr(cos)677 2487 y Fj(\022)p 677 2495 V 677 2524 a Fm(2)716 2507 y Fr(sin)f Fn(\036)642 2567 y Fr(0)602 2627 y(sin)684 2608 y Fj(\022)p 684 2616 V 684 2645 a Fm(2)829 2414 y Fi(1)829 2487 y(C)829 2512 y(C)829 2537 y(C)829 2564 y(A)874 2537 y Fn(;)951 2753 y Fr(49)p eop %%Page: 50 50 50 49 bop 122 56 a Fr(\010)157 36 y Ff(\000)157 69 y(A)p Fj(;)p Fm(2)215 56 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 -67 y Fi(0)443 6 y(B)443 31 y(B)443 56 y(B)443 83 y(@)488 -34 y Fk(\000)p Fn(e)550 -52 y Fj(i\026)601 -34 y Fr(cos)688 -54 y Fj(\022)p 688 -46 18 2 v 688 -17 a Fm(2)727 -34 y Fr(sin)17 b Fn(\036)490 26 y(e)513 8 y Ff(\000)p Fj(i!)594 26 y Fr(cos)681 7 y Fj(\022)p 680 15 V 680 43 a Fm(2)720 26 y Fr(cos)g Fn(\036)584 86 y Fk(\000)8 b Fr(sin)713 67 y Fj(\022)p 713 75 V 713 104 a Fm(2)648 147 y Fr(0)841 -67 y Fi(1)841 6 y(C)841 31 y(C)841 56 y(C)841 83 y(A)885 56 y Fn(;)126 303 y Fr(\010)161 283 y Ff(\000)161 316 y(B)q Fj(;)p Fm(1)215 303 y Fr(\()p Fn(x;)g(\017)p Fr(\))41 b(=)443 180 y Fi(0)443 253 y(B)443 278 y(B)443 303 y(B)443 330 y(@)582 213 y Fk(\000)8 b Fr(sin)711 193 y Fj(\022)p 711 201 V 711 230 a Fm(2)646 273 y Fr(0)491 333 y Fn(e)514 315 y Ff(\000)p Fj(i\026)593 333 y Fr(cos)680 314 y Fj(\022)p 680 322 V 680 351 a Fm(2)719 333 y Fr(sin)17 b Fn(\036)488 394 y(e)511 375 y Ff(\000)p Fj(i!)592 394 y Fr(cos)679 374 y Fj(\022)p 678 382 V 678 411 a Fm(2)718 394 y Fr(cos)g Fn(\036)837 180 y Fi(1)837 253 y(C)837 278 y(C)837 303 y(C)837 330 y(A)881 303 y Fn(;)126 550 y Fr(\010)161 530 y Ff(\000)161 563 y(B)q Fj(;)p Fm(2)215 550 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 427 y Fi(0)443 500 y(B)443 525 y(B)443 550 y(B)443 577 y(@)652 460 y Fr(0)588 520 y Fk(\000)8 b Fr(sin)717 501 y Fj(\022)p 717 508 V 717 537 a Fm(2)488 580 y Fk(\000)p Fn(e)550 562 y Fj(i!)603 580 y Fr(cos)690 561 y Fj(\022)p 690 569 V 690 597 a Fm(2)729 580 y Fr(cos)17 b Fn(\036)511 641 y(e)534 622 y Fj(i\026)585 641 y Fr(cos)672 621 y Fj(\022)p 672 629 V 672 658 a Fm(2)711 641 y Fr(sin)g Fn(\036)848 427 y Fi(1)848 500 y(C)848 525 y(C)848 550 y(C)848 577 y(A)893 550 y Fn(;)0 731 y Fr(for)g Fn(\031)r(=)p Fr(2)d Fn(<)f(\022)i Fk(\024)f Fn(\031)r Fr(,)h(and)122 912 y(\010)157 891 y Fm(+)157 924 y Ff(A)p Fj(;)p Fm(1)215 912 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))41 b(=)443 789 y Fi(0)443 862 y(B)443 887 y(B)443 912 y(B)443 939 y(@)601 822 y Fr(cos)688 802 y Fj(\022)p 687 810 V 687 839 a Fm(2)643 882 y Fr(0)491 942 y Fn(e)514 924 y Ff(\000)p Fj(i\026)593 942 y Fr(sin)675 922 y Fj(\022)p 675 930 V 675 959 a Fm(2)714 942 y Fr(sin)16 b Fn(\036)488 1002 y(e)511 984 y Ff(\000)p Fj(i!)592 1002 y Fr(sin)673 983 y Fj(\022)p 673 991 V 673 1019 a Fm(2)712 1002 y Fr(cos)h Fn(\036)831 789 y Fi(1)831 862 y(C)831 887 y(C)831 912 y(C)831 939 y(A)876 912 y Fn(;)122 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b(=)i Fn( )515 2415 y Fm(4)534 2408 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)122 2481 y Fr(\010)157 2461 y Ff(\000)157 2493 y(A)p Fj(;)p Fm(2)215 2481 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2488 y Fm(3)573 2481 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 2554 y Fr(\010)161 2533 y Ff(\000)161 2566 y(B)q Fj(;)p Fm(1)215 2554 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2561 y Fm(1)573 2554 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)126 2626 y Fr(\010)161 2606 y Ff(\000)161 2639 y(B)q Fj(;)p Fm(2)215 2626 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))40 b(=)i Fk(\000)p Fn( )554 2633 y Fm(2)573 2626 y Fr(\()p Fn(x;)8 b(\017)p Fr(\))i(+)h Fk(O)q Fr(\()p Fn(\026)p Fr(\()p Fn(\017;)d(t)p Fr(\)\))p Fn(;)951 2753 y Fr(50)p eop %%Page: 51 51 51 50 bop 0 -22 a Fl(and)18 b(for)e 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1444 69 2 v 33 x Fj(b)968 1482 y Fd(1)985 1477 y Fj(\021)1004 1465 y Fd(0)1003 1488 y(1)1027 1450 y Fr(\()p Fn(b)1067 1457 y Fm(1)1087 1450 y Fn(\021)1113 1432 y Fm(0)1111 1462 y(1)1132 1450 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1317 1389 y Fi(\023)1058 1549 y Fr(0)613 1613 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 1618 y Fd(2)771 1613 y Fj(y)788 1618 y Fd(2)806 1613 y Ff(\000)p Fj(id)863 1618 y Fd(3)881 1613 y Fj(y)898 1618 y Fd(3)916 1613 y Fm(\))p 613 1624 317 2 v 707 1663 a(2)725 1630 y Fk(p)p 767 1630 69 2 v 33 x Fj(b)782 1668 y Fd(1)799 1663 y Fj(\021)818 1651 y Fd(0)817 1674 y(1)935 1636 y Fn(D)975 1643 y Fj(p)993 1649 y Fh(l)1005 1643 y Ff(\000)p Fm(1)1060 1575 y Fi(\022)1096 1613 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1096 1624 112 2 v 1097 1630 a Fk(p)p 1138 1630 69 2 v 1138 1663 a Fj(b)1153 1668 y Fd(1)1171 1663 y Fj(\021)1190 1651 y Fd(0)1189 1674 y(1)1213 1636 y Fr(\()p Fn(b)1253 1643 y Fm(1)1272 1636 y Fn(\021)1298 1618 y Fm(0)1296 1648 y(1)1318 1636 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1503 1575 y Fi(\023)631 1771 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(e)773 1776 y Fd(4)790 1771 y Fj(y)807 1776 y Fd(4)825 1771 y Ff(\000)p Fj(if)881 1776 y Fd(5)898 1771 y Fm(\))p 631 1782 282 2 v 707 1821 a(2)725 1788 y Fk(p)p 767 1788 69 2 v 33 x Fj(b)782 1826 y Fd(1)799 1821 y Fj(\021)818 1810 y Fd(0)817 1832 y(1)917 1794 y Fn(D)957 1801 y Fj(p)975 1807 y Fh(l)987 1801 y Ff(\000)p Fm(1)1043 1733 y Fi(\022)1078 1771 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1078 1782 112 2 v 1079 1788 a Fk(p)p 1121 1788 69 2 v 33 x Fj(b)1136 1826 y Fd(1)1153 1821 y Fj(\021)1172 1810 y Fd(0)1171 1832 y(1)1195 1794 y Fr(\()p Fn(b)1235 1801 y Fm(1)1255 1794 y Fn(\021)1281 1776 y Fm(0)1279 1806 y(1)1300 1794 y Fn(s)g Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1485 1733 y Fi(\023)1541 1402 y(1)1541 1475 y(C)1541 1500 y(C)1541 1524 y(C)1541 1549 y(C)1541 1574 y(C)1541 1599 y(C)1541 1624 y(C)1541 1649 y(C)1541 1674 y(C)1541 1699 y(C)1541 1724 y(C)1541 1750 y(A)399 1892 y Fr(+)18 b Fk(O)q Fr(\()p Fn(\017)535 1871 y Fj(p)554 1892 y Fr(\))12 b Fn(;)0 1989 y Fl(and)122 2087 y Fn( )154 2094 y Fm(2)173 2087 y Fr(\()p Fn(x;)c(t)p Fr(\))41 b(=)g Fl(e)422 2066 y Fj(iS)r Fm(\()p Fj(t)p Fm(\))p Fj(=\017)530 2054 y Fd(2)547 2066 y Fm(+)p Fj(i\021)q Fm(\()p Fj(t)p Fm(\)\()p Fj(x)p Ff(\000)p Fj(a)p Fm(\()p Fj(t)p Fm(\)\))p Fj(=\017)813 2054 y Fd(2)838 2087 y Fn(C)873 2094 y Fm(2)893 2087 y Fr(\()p Fn(y)r Fr(\))499 2359 y Fk(\002)564 2137 y Fi(0)564 2210 y(B)564 2235 y(B)564 2260 y(B)564 2285 y(B)564 2310 y(B)564 2335 y(B)564 2359 y(B)564 2384 y(B)564 2409 y(B)564 2434 y(B)564 2459 y(B)564 2486 y(@)1131 2165 y Fr(0)867 2252 y Fn(D)907 2259 y Fj(p)925 2265 y Fh(l)947 2192 y Fi(\022)983 2229 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 983 2240 112 2 v 984 2246 a Fk(p)p 1025 2246 69 2 v 1025 2279 a Fj(b)1040 2284 y Fd(1)1058 2279 y Fj(\021)1077 2268 y Fd(0)1076 2290 y(1)1100 2252 y Fr(\()p Fn(b)1140 2259 y Fm(1)1159 2252 y Fn(\021)1185 2234 y Fm(0)1183 2264 y(1)1205 2252 y Fn(s)11 b Fr(+)g Fn(b)g Fk(\001)g Fn(y)r Fr(\))1390 2192 y Fi(\023)717 2358 y Fm(\(1+)p Fj(i)p Fm(\)\()p Fj(e)832 2363 y Fd(4)849 2358 y Fj(y)866 2363 y Fd(4)884 2358 y Fm(+)p Fj(if)940 2363 y Fd(5)958 2358 y Fm(\))p 717 2370 255 2 v 780 2408 a(2)798 2375 y Fk(p)p 839 2375 69 2 v 839 2408 a Fj(b)854 2413 y Fd(1)872 2408 y Fj(\021)891 2397 y Fd(0)890 2419 y(1)976 2381 y Fn(D)1016 2388 y Fj(p)1034 2394 y Fh(l)1047 2388 y Ff(\000)p Fm(1)1102 2321 y Fi(\022)1138 2358 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1138 2370 112 2 v 1139 2375 a Fk(p)p 1180 2375 69 2 v 1180 2408 a Fj(b)1195 2413 y Fd(1)1212 2408 y Fj(\021)1231 2397 y Fd(0)1230 2419 y(1)1255 2381 y Fr(\()p Fn(b)1295 2388 y Fm(1)1314 2381 y Fn(\021)1340 2363 y Fm(0)1338 2393 y(1)1360 2381 y Fn(s)g Fr(+)g Fn(b)f Fk(\001)h Fn(y)r Fr(\))1544 2321 y Fi(\023)613 2487 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\)\()p Fj(c)754 2492 y Fd(2)771 2487 y Fj(y)788 2492 y Fd(2)806 2487 y Fm(+)p Fj(id)863 2492 y Fd(3)881 2487 y Fj(y)898 2492 y Fd(3)916 2487 y Fm(\))p 613 2499 317 2 v 707 2537 a(2)725 2504 y Fk(p)p 767 2504 69 2 v 33 x Fj(b)782 2542 y Fd(1)799 2537 y Fj(\021)818 2526 y Fd(0)817 2548 y(1)935 2510 y Fn(D)975 2517 y Fj(p)993 2523 y Fh(l)1005 2517 y Ff(\000)p Fm(1)1060 2450 y Fi(\022)1096 2487 y Fm(\()p Ff(\000)p Fm(1)p Ff(\000)p Fj(i)p Fm(\))p 1096 2499 112 2 v 1097 2504 a Fk(p)p 1138 2504 69 2 v 1138 2537 a Fj(b)1153 2542 y Fd(1)1171 2537 y Fj(\021)1190 2526 y Fd(0)1189 2548 y(1)1213 2510 y Fr(\()p Fn(b)1253 2517 y Fm(1)1272 2510 y Fn(\021)1298 2492 y Fm(0)1296 2522 y(1)1318 2510 y Fn(s)g Fr(+)g Fn(b)1422 2517 y Fm(1)1441 2510 y Fn(y)1465 2517 y Fm(1)1496 2510 y Fr(+)g Fn(b)1566 2517 y Fm(2)1585 2510 y Fn(y)1609 2517 y Fm(2)1629 2510 y Fr(\))1648 2450 y Fi(\023)1687 2137 y(1)1687 2210 y(C)1687 2235 y(C)1687 2260 y(C)1687 2285 y(C)1687 2310 y(C)1687 2335 y(C)1687 2359 y(C)1687 2384 y(C)1687 2409 y(C)1687 2434 y(C)1687 2459 y(C)1687 2486 y(A)399 2629 y Fr(+)18 b Fk(O)q Fr(\()p Fn(\017)535 2608 y Fj(p)554 2629 y Fr(\))12 b Fn(;)951 2753 y Fr(51)p eop %%Page: 52 52 52 51 bop 0 -22 a Fl(wher)n(e)18 b Fn(C)173 -15 y Fm(2)210 -22 y Fl(is)f(de\014ne)n(d)h(by)143 79 y Fn(C)178 86 y Fm(2)198 79 y Fr(\()p Fn(y)r Fr(\))64 182 y(=)41 b Fk(\000)8 b Fn(\017)210 162 y Ff(\000)p Fj(n=)p Fm(2)305 182 y Fn(')337 189 y Fj(l)349 182 y Fr(\()p Fn(A)405 189 y Fm(0)425 182 y Fn(;)g(B)484 189 y Fm(0)503 182 y Fn(;)g Fr(1)p Fn(;)g Fr(0)p Fn(;)g Fr(0)p Fn(;)g(y)r Fr(\))17 b(exp)816 109 y Fi( )854 149 y Fk(\000)p Fn(\031)r Fr(\()p Fn(c)963 131 y Fm(2)963 161 y(2)982 149 y Fn(y)1008 131 y Fm(2)1006 161 y(2)1038 149 y Fr(+)11 b Fn(d)1112 131 y Fm(2)1112 161 y(3)1132 149 y Fn(y)1158 131 y Fm(2)1156 161 y(3)1189 149 y Fr(+)g Fn(e)1261 131 y Fm(2)1261 161 y(4)1280 149 y Fn(y)1306 131 y Fm(2)1304 161 y(4)1336 149 y Fr(+)g Fn(f)1414 131 y Fm(2)1409 161 y(5)1434 149 y Fr(\))p 854 171 600 2 v 1099 216 a(8)p Fn(b)1144 223 y Fm(1)1163 216 y Fn(\021)1189 199 y Fm(0)1187 227 y(1)1467 109 y Fi(!)143 323 y Fk(\002)26 b Fr(exp)291 250 y Fi( )323 323 y Fn(i)345 290 y(S)378 272 y Ff(B)375 302 y Fm(0)404 290 y Fr(\()p Fn(\017;)8 b Fk(\000)p Fr(\))p 345 312 178 2 v 414 358 a Fn(\017)434 343 y Fm(2)538 323 y Fk(\000)j Fn(i)628 290 y(f)657 272 y Fm(2)652 302 y(5)p 610 312 87 2 v 610 358 a Fn(b)631 365 y Fm(1)650 358 y Fn(\021)676 340 y Fm(0)674 369 y(1)717 323 y Fr(ln)17 b Fn(\017)i Fr(+)11 b Fn(i)915 290 y(f)944 272 y Fm(2)939 302 y(5)p 885 312 111 2 v 885 358 a Fr(4)p Fn(b)930 365 y Fm(1)949 358 y Fn(\021)975 340 y Fm(0)973 369 y(1)1011 323 y Fk(\000)g Fn(i)1113 290 y(f)1142 272 y Fm(2)1137 302 y(5)p 1083 312 V 1083 358 a Fr(2)p Fn(b)1128 365 y Fm(1)1147 358 y Fn(\021)1173 340 y Fm(0)1171 369 y(1)1214 323 y Fr(ln)16 b Fk(j)p Fn(f)1309 330 y Fm(5)1329 323 y Fk(j)j Fr(+)11 b Fn(i)1451 290 y Fr(3)p Fn(f)1504 272 y Fm(2)1499 302 y(5)p 1433 312 V 1433 358 a Fr(4)p Fn(b)1478 365 y Fm(1)1498 358 y Fn(\021)1524 340 y Fm(0)1522 369 y(1)1565 323 y Fr(ln)d(\(2)p Fn(b)1678 330 y Fm(1)1698 323 y Fn(\021)1724 303 y Fm(0)1722 336 y(1)1743 323 y Fr(\))1762 250 y Fi(!)143 465 y Fk(\002)26 b Fr(exp)291 392 y Fi( )332 465 y Fk(\000)p Fn(i)393 431 y(c)414 413 y Fm(2)414 443 y(2)433 431 y Fn(y)459 413 y Fm(2)457 443 y(2)489 431 y Fr(+)11 b Fn(d)563 413 y Fm(2)563 443 y(3)583 431 y Fn(y)609 413 y Fm(2)607 443 y(3)640 431 y Fr(+)g Fn(e)712 413 y Fm(2)712 443 y(4)731 431 y Fn(y)757 413 y Fm(2)755 443 y(4)p 393 453 385 2 v 529 499 a Fr(4)p Fn(b)574 506 y Fm(1)594 499 y Fn(\021)620 482 y Fm(0)618 510 y(1)798 465 y Fr(ln)d(\(2)p Fn(b)911 472 y Fm(1)931 465 y Fn(\021)957 444 y Fm(0)955 477 y(1)976 465 y Fr(\))j(+)g Fn(i)1077 431 y(c)1098 413 y Fm(2)1098 443 y(2)1117 431 y Fn(y)1143 413 y Fm(2)1141 443 y(2)1174 431 y Fr(+)g Fn(d)1248 413 y Fm(2)1248 443 y(3)1268 431 y Fn(y)1294 413 y Fm(2)1292 443 y(3)1324 431 y Fr(+)g Fn(e)1396 413 y Fm(2)1396 443 y(4)1416 431 y Fn(y)1442 413 y Fm(2)1440 443 y(4)p 1077 453 V 1214 499 a Fr(2)p Fn(b)1259 506 y Fm(1)1278 499 y Fn(\021)1304 482 y Fm(0)1302 510 y(1)1482 465 y Fr(ln)17 b Fk(j)p Fn(f)1578 472 y Fm(5)1597 465 y Fk(j)j(\000)10 b Fn(i)1702 431 y Fk(j)p Fn(f)1740 438 y Fm(5)1760 431 y Fk(j)p Fn(b)g Fk(\001)h Fn(y)p 1702 453 154 2 v 1736 499 a(b)1757 506 y Fm(1)1776 499 y Fn(\021)1802 482 y Fm(0)1800 510 y(1)1869 392 y Fi(!)1910 465 y Fn(;)0 612 y(p)24 619 y Fj(l)60 612 y Fr(=)22 b Fk(\000)8 b Fn(i)189 586 y Fj(c)204 575 y Fd(2)204 597 y(2)221 586 y Fj(y)239 575 y Fd(2)238 597 y(2)257 586 y Fm(+)p Fj(d)302 575 y Fd(2)302 597 y(3)320 586 y Fj(y)338 575 y Fd(2)337 597 y(3)356 586 y Fm(+)p Fj(e)399 575 y Fd(2)399 597 y(4)417 586 y Fj(y)435 575 y Fd(2)434 597 y(4)453 586 y Fm(+)p Fj(f)501 575 y Fd(2)497 597 y(5)p 188 601 330 2 v 310 630 a Fm(2)p Fj(b)343 635 y Fd(1)360 630 y Fj(\021)379 618 y Fd(0)378 641 y(1)523 612 y Fl(,)17 b Fn(y)f Fr(=)e(\()p Fn(x)c Fk(\000)h Fn(a)p Fr(\()p Fn(t)p Fr(\)\))p Fn(=\017)p Fl(,)17 b(and)g Fn(s)d Fr(=)g Fn(t=\017)p Fl(.)22 b(Final)r(ly,)d(for)d Fn(\017)1493 594 y Fm(1)p Ff(\000)p Fj(\030)1571 612 y Fk(\024)d Fn(t)h Fk(\024)f Fn(T)7 b Fl(,)122 734 y Fn( )154 741 y Fj(j)172 734 y Fr(\()p Fn(x;)h(t)p Fr(\))41 b(=)g Fl(e)420 714 y Fj(iS)455 702 y Fe(A)482 714 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)555 702 y Fd(2)615 693 y Fi(X)591 787 y Ff(j)p Fj(m)p Ff(j\024j)p Fj(l)p Ff(j)716 734 y Fn(d)741 714 y Ff(A)741 747 y Fj(m;j)809 734 y Fn(')841 741 y Fj(m)874 734 y Fr(\()p Fn(A)930 714 y Ff(A)960 734 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1078 714 y Ff(A)1107 734 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1205 714 y Fm(2)1224 734 y Fn(;)g(a)1272 714 y Ff(A)1302 734 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1406 714 y Ff(A)1435 734 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1603 714 y Fm(+)1603 747 y Ff(A)p Fj(;j)1659 734 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 865 y(+)17 b Fl(e)476 845 y Fj(iS)511 833 y Fe(B)534 845 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)607 833 y Fd(2)642 824 y Fi(X)656 911 y Fj(m)719 865 y Fn(d)744 842 y Ff(B)q Fj(;)p Fm(1)744 877 y Fj(m;j)812 865 y Fn(')844 872 y Fj(m)877 865 y Fr(\()p Fn(A)933 845 y Ff(B)958 865 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1076 845 y Ff(B)1101 865 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1199 845 y Fm(2)1218 865 y Fn(;)g(a)1266 845 y Ff(B)1292 865 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1396 845 y Ff(B)1421 865 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1589 845 y Fm(+)1589 878 y Ff(B)q Fj(;)p Fm(1)1642 865 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 981 y(+)17 b Fl(e)476 960 y Fj(iS)511 949 y Fe(B)534 960 y Fm(\()p Fj(t)p Fm(\))p Fj(=\017)607 949 y Fd(2)642 940 y Fi(X)656 1027 y Fj(m)719 981 y Fn(d)744 957 y Ff(B)q Fj(;)p Fm(2)744 992 y Fj(m;j)812 981 y Fn(')844 988 y Fj(m)877 981 y Fr(\()p Fn(A)933 960 y Ff(B)958 981 y Fr(\()p Fn(t)p Fr(\))p Fn(;)8 b(B)1076 960 y Ff(B)1101 981 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\017)1199 960 y Fm(2)1218 981 y Fn(;)g(a)1266 960 y Ff(B)1292 981 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(\021)1396 960 y Ff(B)1421 981 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g(x)p Fr(\))g(\010)1589 960 y Fm(+)1589 993 y Ff(B)q Fj(;)p Fm(2)1642 981 y Fr(\()p Fn(x;)g(t;)g(\017)p Fr(\))398 1085 y(+)26 b Fk(O)q Fr(\()p Fn(\017)542 1064 y Fj(p)561 1085 y Fr(\))p Fn(;)0 1185 y Fl(wher)n(e)21 b(the)f Fn(d)249 1167 y Ff(A)249 1197 y Fj(m;j)329 1185 y Fl(and)h Fn(d)452 1161 y Ff(B)q Fj(;k)452 1196 y(m;j)532 1185 y Fl(ar)n(e)f(c)n(ompute)n(d)g(by)g(the)h(pr)n(o)n(c)n(ess)e(use)n (d)i(in)f(the)h(pr)n(o)n(of)e(of)h(lemma)h(3.6.)31 b(The)0 1245 y(initial)14 b(c)n(onditions)h(for)e(the)h(semiclassic)n(al)h (quantities)g(in)f(the)g(expr)n(ession)g(for)f Fn( )1492 1252 y Fj(j)1510 1245 y Fr(\()p Fn(x;)8 b(t)p Fr(\))13 b Fl(for)g Fn(t)h Fk(\025)f Fn(\017)1807 1227 y Fm(1)p Ff(\000)p Fj(\030)1885 1245 y Fl(ar)n(e)0 1305 y(the)20 b(analo)n(gs)f(of)g(formulas)g(\(3.71\))g(and)g(\(3.72\))f(after)i(one) f(has)g(taken)i(into)e(ac)n(c)n(ount)h(the)f(appr)n(opriate)0 1366 y(alter)n(ations)f(fr)n(om)e(se)n(ction)i(2.7.)0 1532 y Fo(References)156 1641 y Fr([1])24 b(Abromo)o(witz,)15 b(M.)i(and)h(Stegun,)g(I.)f(A.:)47 b Fl(Handb)n(o)n(ok)19 b(of)f(Mathematic)n(al)h(F)l(unctions)p Fr(.)g(New)232 1701 y(Y)l(ork:)i(Do)o(v)o(er)15 b(1968.)156 1803 y([2])24 b(Cole,)i(J.)e(D.:)48 b Fl(Perturb)n(ation)25 b(Metho)n(ds)g(in)g (Applie)n(d)g(Mathematics)g Fr(W)l(altham,)g(Mass.,)232 1863 y(T)l(oron)o(to,)16 b(London:)23 b(Blaisdell)15 b(1968.)156 1964 y([3])24 b(Com)o(b)q(es,)d(J.-M.:)48 b(On)22 b(the)f(Born{Opp)q(enheimer)f(Appro)o(ximation.)f(In:)32 b Fl(International)232 2024 y(Symp)n(osium)19 b(on)i(Mathematic)n(al)g (Pr)n(oblems)g(in)g(The)n(or)n(etic)n(al)f(Physics.)f Fr(ed.)g(b)o(y)g(H.)g(Araki.)232 2084 y(Berlin,)14 b(Heidelb)q(erg,)g (New)i(Y)l(ork:)21 b(Springer)16 b(1975.)156 2185 y([4])24 b(Com)o(b)q(es,)c(J.-M.:)47 b(The)21 b(Born{Opp)q(enheimer)d(Appro)o (ximation.)h(In:)29 b Fl(The)21 b(Schr\177)-25 b(odinger)232 2246 y(Equation.)17 b Fr(ed.)e(b)o(y)h(W.)g(Thirring,)g(P)l(.)g(Urban.) g(Wien,)f(New)h(Y)l(ork:)k(Springer)c(1977.)156 2347 y([5])24 b(Com)o(b)q(es,)18 b(J.-M.,)g(Duclos)g(P)l(.,)h(and)g(Seiler)e (R.:)48 b(The)19 b(Born{Opp)q(enheimer)e(Appro)o(xima-)232 2407 y(tion.)g(In)h Fl(R)o(igor)n(ous)f(A)o(tomic)i(and)g(Mole)n(cular) h(Physics.)d Fr(ed.)h(b)o(y)f(G.)h(V)l(elo,)f(A.)g(Wigh)o(tman.)232 2467 y(New)e(Y)l(ork:)21 b(Plen)o(um)14 b(1981.)156 2568 y([6])24 b(Gradsh)o(teyn,)c(I.)f(S.)h(and)g(Ryzhik,)f(I.)g(M.:)48 b Fl(T)l(able)22 b(of)e(Inte)n(gr)n(als,)i(Series,)g(and)f(Pr)n(o)n (ducts.)232 2629 y Fr(New)15 b(Y)l(ork:)21 b(Academic)14 b(Press)i(1980.)951 2753 y(52)p eop %%Page: 53 53 53 52 bop 156 -22 a Fr([7])24 b(Hagedorn,)15 b(G.)h(A.:)22 b(A)15 b(Time{Dep)q(enden)o(t)e(Born{Opp)q(enheimer)h(Appro)o (ximation.)f Fl(Com-)232 39 y(mun.)k(Math.)h(Phys.)d Fg(77)p Fr(,)h(1{19)i(\(1980\).)156 140 y([8])24 b(Hagedorn,)d(G.)f (A.:)47 b(Semiclassical)18 b(Quan)o(tum)h(Mec)o(hanics)g(IV:)g(Large)i (Order)e(Asymp-)232 200 y(totics)24 b(and)g(More)g(General)g(States)h (in)f(More)g(than)g(One)g(Dimension.)e Fl(A)o(nn.)k(Inst.)f(H.)232 261 y(Poinc)n(ar)o(\023)-24 b(e)17 b(Se)n(ct.)i(A)d Fg(42)p Fr(,)g(363{374)j(\(1985\).)156 362 y([9])24 b(Hagedorn,)36 b(G.)c(A.:)47 b(High)32 b(Order)f(Corrections)h(to)h(the)e(Time{Dep)q (enden)o(t)g(Born{)232 423 y(Opp)q(enheimer)21 b(Appro)o(ximation)h(I:) h(Smo)q(oth)h(P)o(oten)o(tials.)f Fl(A)o(nn.)i(Math.)e Fg(124)p Fr(,)i(571{590)232 483 y(\(1986\).)17 b(Erratum)e Fg(126)p Fr(,)h(219)i(\(1987\).)131 584 y([10])25 b(Hagedorn,)18 b(G.)g(A.:)47 b(High)18 b(Order)g(Corrections)g(to)g(the)g(Time{Indep)q (enden)o(t)d(Born{Op-)232 645 y(p)q(enheimer)g(Appro)o(ximation)g(I:)h (Smo)q(oth)h(P)o(oten)o(tials.)f Fl(A)o(nn.)j(Inst.)g(H.)f(Poinc)n(ar)o (\023)-24 b(e)19 b(Se)n(ct.)g(A)232 705 y Fg(47)p Fr(,)d(1{19)h (\(1987\).)131 807 y([11])25 b(Hagedorn,)18 b(G.)g(A.:)47 b(High)18 b(Order)g(Corrections)g(to)g(the)g(Time{Indep)q(enden)o(t)d (Born{Op-)232 867 y(p)q(enheimer)10 b(Appro)o(ximation)h(I)q(I:)h (Diatomic)f(Coulom)o(b)h(Systems.)f Fl(Commun.)k(Math.)f(Phys.)232 927 y Fg(116)p Fr(,)i(23{44)i(\(1988\).)131 1029 y([12])25 b(Hagedorn,)36 b(G.)c(A.:)47 b(High)32 b(Order)f(Corrections)h(to)h (the)e(Time{Dep)q(enden)o(t)g(Born{)232 1089 y(Opp)q(enheimer)14 b(Appro)o(ximation)h(I)q(I:)h(Coulom)o(b)g(Systems.)f Fl(Commun.)j(Math.)g(Phys.)e Fg(117)p Fr(,)232 1149 y(387{403)j (\(1988\).)131 1251 y([13])25 b(Hagedorn,)57 b(G.)49 b(A.:)f(Multiple)f(Scales)i(and)h(the)f(Time{Indep)q(enden)o(t)e(Born{) 232 1311 y(Opp)q(enheimer)20 b(Appro)o(ximation.)h(In:)34 b Fl(Di\013er)n(ential)23 b(Equations)i(and)e(Applic)n(ations.)h Fr(ed.)232 1371 y(b)o(y)15 b(R.)h(Aftabizadeh.)f(New)h(Y)l(ork:)21 b(Marcel)15 b(Dekk)o(er)g(1989.)131 1473 y([14])25 b(Hagedorn,)k(G.)e (A.:)48 b(Electron)26 b(Energy)h(Lev)o(el)f(Crossings)i(in)f(the)g (Time{Dep)q(enden)o(t)232 1533 y(Born{Opp)q(enheimer)14 b(Appro)o(ximation.)g Fl(The)n(or.)j(Chimic)n(a)f(A)n(cta.)h Fg(77)p Fr(,)f(163{190)j(\(1990\).)131 1635 y([15])25 b(Hagedorn,)e(G.)e(A.:)47 b(Pro)q(of)23 b(of)f(the)f(Landau{Zener)h(F)l (orm)o(ula)e(in)h(an)h(Adiabatic)f(Limit)232 1695 y(with)16 b(Small)e(Eigen)o(v)m(alue)i(Gaps.)h Fl(Commun.)g(Math.)g(Phys.)f Fg(136)p Fr(,)g(433{449)j(\(1991\).)131 1797 y([16])25 b(Hagedorn,)f(G.)f(A.:)47 b(Time{Rev)o(ersal)21 b(In)o(v)m(ariance)h (and)h(the)g(Time{Dep)q(enden)o(t)e(Born{)232 1857 y(Opp)q(enheimer)g (Appro)o(ximation.)h(In)i Fl(F)l(orty)f(Mor)n(e)h(Y)l(e)n(ars)g(of)g(R) n(ami\014c)n(ations:)36 b(Sp)n(e)n(ctr)n(al)232 1917 y(Asymptotics)20 b(and)h(Its)f(Applic)n(ations,)h Fr(\(Discourses)f(in) f(Mathematics)f(and)i(Its)f(Applica-)232 1977 y(tions,)c(No.)g(1\).)h (ed.)f(b)o(y)g(S.)g(A.)g(F)l(ulling)g(and)h(F.)f(J.)g(Narco)o(wic)o(h.) f(College)h(Station:)22 b(T)l(exas)15 b(A)232 2037 y(&)h(M)g(Univ)o (ersit)o(y)d(Mathematics)i(Departmen)o(t)g(1992.)131 2139 y([17])25 b(Hagedorn,)15 b(G.)g(A.:)47 b(Molecular)14 b(Propagation)j(Through)f(Electronic)d(Eigen)o(v)m(alue)i(Cross-)232 2199 y(ings,)h Fl(Memoirs)g(A)o(mer.)i(Math.)f(So)n(c.)f Fg(536)p Fr(,)g(\(1994\).)131 2301 y([18])25 b(Hagedorn,)13 b(G.)g(A.:)47 b(E\013ects)13 b(of)g(Electron)g(Energy)f(Lev)o(el)g (Crossings)i(on)f(Molecular)f(Prop-)232 2361 y(agation.)27 b Fl(Di\013er)n(ential)g(Equations)h(and)f(Mathematic)n(al)g(Physics.)g (Pr)n(o)n(c)n(e)n(e)n(dings)f(of)g(the)232 2421 y(International)d (Confer)n(enc)n(e.)f(University)g(of)f(A)o(lab)n(ama)g(at)h(Birmingham) f(Mar)n(ch)g(13{17,)232 2482 y(1994.)16 b Fr(ed.)f(b)o(y)h(I.)f(Kno)o (wles,)h(85{95,)i(1995.)951 2753 y(53)p eop %%Page: 54 54 54 53 bop 131 -22 a Fr([19])25 b(Hagedorn,)g(G.)e(A.:)48 b(Classi\014cation)23 b(and)h(Normal)e(F)l(orms)g(for)i(Av)o(oided)e (Crossings)j(of)232 39 y(Quan)o(tum)12 b(Mec)o(hanical)h(Energy)g(Lev)o (els.)g(Preprin)o(t,)g(Virginia)g(P)o(olytec)o(hnic)e(Institute)i(and) 232 99 y(State)j(Univ)o(ersit)o(y)l(.)131 200 y([20])25 b(Hagedorn,)17 b(G.)f(A.,)g(and)h(Jo)o(y)o(e,)e(A.:)48 b(Landau{Zener)18 b(T)l(ransitions)f(through)h(Small)d(Elec-)232 261 y(tronic)j(Gaps)i(in)e(the)h(Born{Opp)q(enheimer)e(Appro)o (ximation.)f(Preprin)o(t,)i(Virginia)g(P)o(oly-)232 321 y(tec)o(hnic)c(Institute)i(and)h(State)f(Univ)o(ersit)o(y)l(.)131 423 y([21])25 b(Herrin,)15 b(J.)h(S.:)48 b(The)17 b(Born{Opp)q (enheimer)e(Appro)o(ximation:)g(Straigh)o(t{Up)i(and)g(with)g(a)232 483 y(Twist.)f(Ph.)g(D.)g(Dissertation,)g(Univ)o(ersit)o(y)e(of)i (Virginia,)f(1990.)131 584 y([22])25 b(Jo)o(y)o(e,)20 b(A.:)47 b(Non-trivial)20 b(Prefactors)g(in)h(Adiabatic)e(T)l (ransition)i(Probabilities)f(Induced)232 645 y(b)o(y)15 b(High)h(Order)g(Complex)f(Degeneracies.)g Fl(J.)i(Phys.)g(A)f Fg(26)p Fr(,)g(6517{6540)k(\(1993\).)131 746 y([23])25 b(Jo)o(y)o(e,)15 b(A.:)48 b(Pro)q(of)17 b(of)g(the)g(Landau{Zener)h(F)l (orm)o(ula.)c Fl(Asymptotic)k(A)o(nalysis)f Fg(9)p Fr(,)f(209{258)232 807 y(\(1994\).)131 908 y([24])25 b(Jo)o(y)o(e,)20 b(A.:)47 b(Exp)q(onen)o(tial)20 b(Asymptotics)f(in)h(a)h(Singular)f(Limit)f(for) i Fn(n)p Fr(-Lev)o(el)e(Scattering)232 968 y(Systems.)h(Preprin)o(t)i (CNRS)g(Marseille)e(CPT{95/P)l(.321)q(6,)26 b Fl(SIAM)d(J.)g(Math.)g(A) o(nal.)g Fr(\(to)232 1029 y(app)q(ear\).)131 1130 y([25])i(Jo)o(y)o(e,) 18 b(A.,)h(Kunz,)g(H.,)f(and)i(P\014ster,)f(C.-E.:)48 b(Exp)q(onen)o(tial)19 b(Deca)o(y)f(and)i(Geometric)d(As-)232 1191 y(p)q(ect)d(of)h(T)l(ransition)h(Probabilities)e(in)g(the)h (Adiabatic)f(Limit.)e Fl(A)o(nn.)17 b(Phys.)e Fg(208)p Fr(,)f(299{332)232 1251 y(\(1991\).)131 1352 y([26])25 b(Jo)o(y)o(e,)c(A.,)h(Mileti,)e(G.,)i(and)h(P\014ster,)f(C.-E.:)48 b(In)o(terferences)20 b(in)h(Adiabatic)g(T)l(ransition)232 1413 y(Probabilities)15 b(Mediated)h(b)o(y)f(Stok)o(es)h(Lines.)g Fl(Phys.)h(R)n(ev.)g(A)g Fg(44)p Fr(,)f(4280{4295)j(\(1991\).)131 1514 y([27])25 b(Jo)o(y)o(e,)15 b(A.)h(and)h(P\014ster,)f(C.-E.:)48 b(F)l(ull)15 b(Asymptotic)g(Expansion)i(of)g(T)l(ransition)g(Probabil-) 232 1575 y(ities)e(in)h(the)g(Adiabatic)f(Limit.)f Fl(J.)j(Phys.)g(A.)g Fg(24)p Fr(,)e(753{766)k(\(1991\).)131 1676 y([28])25 b(Jo)o(y)o(e,)16 b(A.)g(and)h(P\014ster,)g(C.-E.:)48 b(Absence)16 b(of)h(Geometrical)e(Correction)i(to)g(the)g(Landau-)232 1736 y(Zener)e(F)l(orm)o(ula.)f Fl(Phys.)j(L)n(ett.)h(A)e Fg(169)p Fr(,)g(62{66)i(\(1992\).)131 1838 y([29])25 b(Jo)o(y)o(e,)14 b(A.)h(and)h(P\014ster,)g(C.-E.:)48 b(Sup)q(eradiabatic)16 b(Ev)o(olution)f(and)i(Adiabatic)e(T)l (ransition)232 1898 y(Probabilit)o(y)21 b(b)q(et)o(w)o(een)g(Tw)o(o)i (Non{Degenerate)f(Lev)o(els)f(Isolated)h(in)f(the)h(Sp)q(ectrum.)e Fl(J.)232 1958 y(Math.)d(Phys.)e Fg(34)p Fr(,)h(454{479)j(\(1993\).)131 2060 y([30])25 b(Jo)o(y)o(e,)11 b(A.)f(and)i(P\014ster,)g(C.-E.:)48 b(Non-ab)q(elian)12 b(Geometric)d(E\013ect)i(in)g(Quan)o(tum)g (Adiabatic)232 2120 y(T)l(ransitions.)16 b Fl(Phys.)h(R)n(ev.)g(A)g Fg(48)p Fr(,)f(2598{2608)j(\(1993\).)131 2222 y([31])25 b(Jo)o(y)o(e,)18 b(A.)g(and)i(P\014ster,)f(C.-E.:)48 b(Quan)o(tum)18 b(Adiabatic)g(Ev)o(olution,)h(in)g Fl(L)n(euven)h (Confer-)232 2282 y(enc)n(e)i(Pr)n(o)n(c)n(e)n(e)n(dings;)f(On)h(the)f (Thr)n(e)n(e)f(L)n(evels)i(Micr)n(o-)e(Meso-)h(and)g(Macr)n(o-Appr)n(o) n(aches)f(in)232 2342 y(Physics)p Fr(,)j(M.)e(F)l(annes,)j(C.)e(Meas,)h (A.)e(V)l(erb)q(eure)g(edts.,)i(Plen)o(um,)e(New)h(Y)l(ork,)g(139-148) 232 2403 y(\(1994\).)131 2504 y([32])j(Jo)o(y)o(e,)12 b(A.,)h(P\014ster,)g(C.-E.)g(:)48 b(Semi-Classical)12 b(Asymptotics)f(b)q(ey)o(ond)i(All)f(Orders)h(for)g(Sim-)232 2565 y(ple)i(Scattering)h(Systems,)f Fl(SIAM)i(J.Math.A)o(nal.)f Fg(26)p Fr(,)g(944{977)j(\(1995\).)951 2753 y(54)p eop %%Page: 55 55 55 54 bop 131 -22 a Fr([33])25 b(Kargol,)14 b(A.:)k(The)13 b(In\014nite)g(Time)e(Limit)g(for)j(the)e(Time{Dep)q(enden)o(t)g (Born{Opp)q(enheimer)232 39 y(Appro)o(ximation.)i Fl(Commun.)j(Math.)g (Phys.)f Fg(166)p Fr(,)g(129{148)j(\(1994\).)131 140 y([34])25 b(Kato,)13 b(T.:)48 b(On)13 b(the)g(Adiabatic)f(Theorem)f(in) i(Quan)o(tum)e(Mec)o(hanics.)g Fl(J.)j(Phys.)g(So)n(c.)g(Jpn.)232 200 y Fg(5)p Fr(,)i(435{439)i(\(1950\).)131 302 y([35])25 b(Klein,)c(M.:)48 b(On)21 b(the)h(Mathematical)d(Theory)j(of)g (Predisso)q(ciation.)f Fl(A)o(nn.)i(Phys.)e Fg(178)p Fr(,)232 362 y(48{73)d(\(1987\).)131 464 y([36])25 b(Klein,)12 b(M.,)h(Martinez,)g(A.,)g(Seiler)f(R.,)i(and)g(W)l(ang,)h(X.)d(P)l(.:) 48 b(On)14 b(the)g(Born{Opp)q(enheimer)232 524 y(Expansion)f(for)f(P)o (oly)o(atomic)e(Molecules.)g Fl(Commun.)k(Math.)f(Phys.)f Fg(143)p Fr(,)g(607{639)j(\(1992\).)131 626 y([37])25 b(Klein,)14 b(M.,)h(Martinez,)f(A.,)h(and)h(W)l(ang,)h(X.)e(P)l(.:)48 b(On)16 b(the)f(Born{Opp)q(enheimer)f(Appro)o(x-)232 686 y(imation)f(of)i(W)l(a)o(v)o(e)f(Op)q(erators)i(in)e(Molecular)g (Scattering)g(Theory)l(.)h(Univ)o(ersit)o(\023)-23 b(e)12 b(de)i(Nan)o(tes)232 746 y(preprin)o(t,)h(1992.)131 848 y([38])25 b(Martin,)j(Ph.A.)d(and)i(Nenciu)e(G.:)89 b(Semi-Classical)25 b(Inelastic)g Fn(S)s Fr(-Matrix)h(for)g(One-)232 908 y(Dimensional)14 b Fn(N)5 b Fr(-States)17 b(Systems,)e Fl(R)n(ev.)i(Math.)g(Phys.)f Fg(7)p Fr(,)g(193{242)j(\(1995\).)131 1010 y([39])25 b(Martinez,)232 1070 y(A.:)47 b(D)o(\023)-23 b(ev)o(elopp)q(emen)o(ts)20 b(Asymptotiques)g(et)i(E\013et)g(T)l(unnel) g(dans)h(l'Appro)o(ximation)c(de)232 1130 y(Born{Opp)q(enheimer.)14 b Fl(A)o(nn.)k(Inst.)g(H.)f(Poinc)n(ar)o(\023)-24 b(e)18 b(Se)n(ct.)g(A)e Fg(50)p Fr(,)g(239{257)j(\(1989\).)131 1232 y([40])25 b(Martinez,)17 b(A.:)48 b(D)o(\023)-23 b(ev)o(elopp)q(emen)o(ts)16 b(Asymptotiques)g(dans)k(l'Appro)o(xim)o (ation)c(de)j(Born{)232 1292 y(Opp)q(enheimer.)13 b Fl(Journ)o(\023)-24 b(ees)18 b(E.)g(D.)f(P.)g(de)h(St.)g(Je)n(an-de-Monts)f Fr(\(1988\).)131 1394 y([41])25 b(Martinez,)g(A.:)48 b(Resonances)25 b(dans)h(l'Appro)o(ximation)c(de)j(Born{Opp)q(enheimer) e(I.)h Fl(J.)232 1454 y(Di\013.)17 b(Eq.)f Fg(91)p Fr(,)g(204{234)j (\(1991\).)131 1556 y([42])25 b(Martinez,)35 b(A.:)47 b(Resonances)33 b(dans)g(l'Appro)o(ximation)d(de)i(Born{Opp)q(enheimer) f(I)q(I.)232 1616 y(Largeur)17 b(de)f(R)o(\023)-23 b(esonances.)16 b Fl(Commun.)i(Math.)f(Phys.)e Fg(135)p Fr(,)h(517{530)j(\(1991\).)131 1718 y([43])25 b(Reed,)11 b(M.)g(and)h(Simon,)f(B.:)47 b Fl(Metho)n(ds)13 b(of)g(Mo)n(dern)f(Mathematic)n(al)h(Physics)g(I:)g (F)l(unctional)232 1778 y(A)o(nalysis)p Fr(.)j(New)f(Y)l(ork,)h (London:)23 b(Academic)13 b(Press)k(1972.)951 2753 y(55)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF