Content-Type: multipart/mixed; boundary="-------------0607241637508" This is a multi-part message in MIME format. ---------------0607241637508 Content-Type: text/plain; name="06-208.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-208.keywords" Resonances; Hilbert sapce; rigged Hilbert space; Gamow states ---------------0607241637508 Content-Type: text/plain; name="aip-6s.clo" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="aip-6s.clo" %% %% This is file `aip-6s.clo', %% generated with the docstrip utility. %% %% The original source files were: %% %% aipparms.dtx (with options: `aip6x9single') %% %% Layout file for class aipproc to use with LaTeX2e %% (C) 2000 American Institute of Physics and Frank Mittelbach %% All rights reserved %% %% $Id: aipparms.dtx,v 1.20 2005/12/01 16:16:27 frank Exp $ \setlength\bodytextsize {12pt} \setlength\bodytextbaselineskip {13.6pt} \setlength\bodytextenspace {5pt} \setlength\bodytextparindent {12pt} \SetInternalRegister\@lowpenalty {51} \SetInternalRegister\@medpenalty {151} \SetInternalRegister\@highpenalty {301} \renewcommand\normalsize{ \@setfontsize\normalsize{\bodytextsize}{\bodytextbaselineskip} \setlength\abovedisplayskip {\bodytextsize plus 2pt minus 5pt} \setlength\belowdisplayskip {\abovedisplayskip} \setlength\abovedisplayshortskip {0pt plus 3pt} \setlength\belowdisplayshortskip {.6\bodytextsize plus 3pt minus 3pt} \let\@listi\@listI } \renewcommand\small{\@setfontsize\small{11pt}{12.8pt} \setlength\abovedisplayskip {11pt plus 3pt minus 6pt} \setlength\belowdisplayskip {\abovedisplayskip} \setlength\abovedisplayshortskip {0pt plus 3pt} \setlength\belowdisplayshortskip {6.5pt plus 3.5pt minus 3pt} \let\@listi\@listIsmall } \renewcommand\footnotesize{\@setfontsize\footnotesize{10}{11} \setlength\abovedisplayskip {10pt plus 2pt minus 5pt} \setlength\belowdisplayskip {\abovedisplayskip} \setlength\abovedisplayshortskip {0pt plus 3pt} \setlength\belowdisplayshortskip {6pt plus 3pt minus 3pt} \let\@listi\@listIfootnotesize } \renewcommand\tiny{\@setfontsize\tiny{5}{6}} \AIP@cmdnotsupported\scriptsize \AIP@cmdnotsupported\large \AIP@cmdnotsupported\Large \AIP@cmdnotsupported\LARGE \AIP@cmdnotsupported\huge \AIP@cmdnotsupported\Huge \ifthenelse{\boolean{@cmrfonts}}{} { \renewcommand\sfdefault{phv} \renewcommand\rmdefault{ptm} \renewcommand\ttdefault{pcr} \renewcommand\bfdefault{b} } \SetInternalRegister\thinmuskip {2mu} \SetInternalRegister\medmuskip {2.5mu plus 1mu minus 1mu} \SetInternalRegister\thickmuskip{4mu plus 1.5mu minus 1mu} \DeclareParagraphLayout{AIPbodytext} {\bodytextsize}{\bodytextbaselineskip} {\bodytextparindent}{0pt}{0pt}{0pt plus 1fil} {0pt} {} \SetInternalRegister\tolerance{500} \SetInternalRegister\pretolerance{300} \setlength\emergencystretch{2pc} \frenchspacing \setlength\textwidth {35pc} \setlength\columnsep {0pt} \setlength\topskip {12pt} \setlength\textheight{\bodytextbaselineskip * 45 + \topskip} \setlength\headheight {\bodytextbaselineskip} \setlength\headsep {2pc} \setlength\footskip {2pc} \setlength\topmargin {(\paperheight - \textheight ) / 2 - 1in - \headheight - \headsep } \setlength\oddsidemargin { (\paperwidth - \textwidth)/2 -1in } \setlength\evensidemargin { \paperwidth -2in - \textwidth - \oddsidemargin} \SetInternalRegister\clubpenalty {9999} \SetInternalRegister\widowpenalty {3000} \SetInternalRegister\displaywidowpenalty {50} \SetInternalRegister\brokenpenalty {100} \SetInternalRegister\predisplaypenalty {10000} \SetInternalRegister\postdisplaypenalty {0} \raggedbottom \newcommand\XFMtitleblock{ \XFMtitle \XFMauthors \XFMaddresses \XFMabstract \vspace{5pt} \XFMkeywords \XFMclassification } \newcommand\XFMtitleformat[1] {#1} \DeclareParagraphLayout{XFMtitle} {20pt}{21pt} {0pt}{\fill}{\fill}{0pt} {0pt} {\SetInternalRegister\hyphenpenalty{2000} \SetInternalRegister\finalhyphendemerits{20000} \bfseries \let\\\@centercr } \setlength\XFMtitleBBskip {0.5in} \DeclareParagraphLayout{XFMauthors} {14pt}{15pt} {0pt}{\fill}{\fill}{0pt} {0pt} {\SetInternalRegister\hyphenpenalty{2000} \SetInternalRegister\finalhyphendemerits{20000} \normalfont \let\\\@centercr } \newcommand\XFMauthorscommatext {,~} \newcommand\XFMauthorsandtext {~ and~} \newcommand\XFMauthorsandtwotext {~ and~} \newcommand\XFMauthorsaddressmarkfont {\fontsize{9}{9}\selectfont} \newcommand\XFMauthorsaddressmarkformat[1] {\textsuperscript{#1}} \setlength\XFMauthorsBBskip {34pt - 6pt} \DeclareParagraphLayout{XFMaddress} {10pt}{11pt} {0pt}{\bodytextparindent plus 1fil} {\bodytextparindent plus 1fil}{0pt} {0pt} {\SetInternalRegister\hyphenpenalty{2000} \SetInternalRegister\finalhyphendemerits{20000} \itshape \let\\\@centercr } \newcommand\XFMaddressmarkstyle {\fnsymbol} \newcommand\XFMaddressmarkfont {\fontsize{7}{7}\selectfont} \newcommand\XFMaddressmarkformat[1] {\textsuperscript{#1}} \newcommand\XFMauthorsaltaddressmarkseparator{,} \newcommand\XFMaddressseparator {\par} \setlength\XFMaddressBBskip {28pt - 4pt} \renewcommand\abstractname {Abstract} \newcommand\XFMabstractheadingfont {\fontsize{10}{11}\bfseries} \newcommand\XFMabstractheadingformat[1] {#1.\hspace{.5em}} \DeclareParagraphLayout{XFMabstracttext} {10pt}{11pt} {1pc}{0pt}{0pt}{0pt plus 1fil} {0pt} {\normalfont} \setlength\XFMabstractBBskip {28pt} \setlength\XFMabstractwidth {\textwidth -2\bodytextparindent} \setlength\XFMabstractleftindent{\bodytextparindent} \DeclareParagraphLayout{XFMclassificationtext} {10pt}{11pt} {0pt}{\bodytextparindent}{\bodytextparindent}{0pt plus 1fil} {0pt} {\normalfont} \setlength\XFMclassificationBBskip {11pt} \newcommand\classificationname {PACS} \newcommand\XFMclassificationheadingfont {\bfseries} \newcommand\XFMclassificationheadingformat[1] {#1:\hspace{.5em}} \classification{\AIP@error{Missing~ \noexpand\classification declaration}{Specify~ PACS~ number(s)~ choosing~ from~ http://www.aip.org/pacs/index.html} } \newcommand\keywordsname {Keywords} \newcommand\XFMkeywordsheadingfont {\bfseries} \newcommand\XFMkeywordsheadingformat[1] {#1:\hspace{.5em}} \DeclareParagraphLayoutAlias{XFMkeywordstext}{XFMclassificationtext} \keywords{\AIP@error{Missing~ \noexpand\keywords declaration} {Specify~ a~ list~ of~ keywords} } \setlength\XFMkeywordsBBskip {11pt} \newcommand\XFMtitleblockpostcode{} \newcommand\XFMtitleblockmarkstyle{\arabic} \setlength\XFMtitleblockpostskip{2pc - 6pt} \renewcommand\thesection {\arabic{section}} \renewcommand\thesubsection {\thesection.\arabic{subsection}} \renewcommand\thesubsubsection {\thesubsection .\arabic{subsubsection}} \renewcommand\theparagraph {\thesubsubsection.\arabic{paragraph}} \renewcommand\@seccntformat[1]{ {\csname the#1\endcsname.\enspace} } \newcommand\AIPsectionpenalty {-1000} \newcommand\AIPsectionindent {0pt} \newcommand\AIPsectionpreskip {2\bodytextbaselineskip plus 3pt minus 1pt} %% actually we use 2pts less than the above: \renewcommand\AIPsectionpreskip {21.4pt plus 3pt minus 1pt} \newcommand\AIPsectionpostskip {\bodytextbaselineskip} \newcommand\AIPsectionafterindent {false} \newcommand\AIPsectionrunin {false} \newcommand\AIPsectionformat[1] {\centering\MakeTextUppercase{#1}} \newcommand\AIPsectionfont {\normalfont\fontsize{14}{16}\bfseries} \newcommand\AIPsubsectionpenalty {-300} \newcommand\AIPsubsectionindent {0pt} \newcommand\AIPsubsectionpreskip {2\bodytextbaselineskip plus 3pt minus 1pt} \newcommand\AIPsubsectionpostskip {\bodytextbaselineskip} \newcommand\AIPsubsectionafterindent {true} \newcommand\AIPsubsectionrunin {false} \newcommand\AIPsubsectionformat[1] {\centering#1} \newcommand\AIPsubsectionfont {\normalfont\fontsize{14}{16}\bfseries} \newcommand\AIPsubsubsectionpenalty {-300} \newcommand\AIPsubsubsectionindent {0pt} \newcommand\AIPsubsubsectionpreskip {2\bodytextbaselineskip plus 3pt minus 1pt} \newcommand\AIPsubsubsectionpostskip {\bodytextbaselineskip} \newcommand\AIPsubsubsectionafterindent {true} \newcommand\AIPsubsubsectionrunin {false} \newcommand\AIPsubsubsectionformat[1] {\centering#1} \newcommand\AIPsubsubsectionfont {\normalfont\fontsize{14}{16}\itshape} \newcommand\AIPparagraphpenalty {-300} \newcommand\AIPparagraphindent {\bodytextparindent} \newcommand\AIPparagraphpreskip {\bodytextbaselineskip plus 3pt minus 1pt} \newcommand\AIPparagraphpostskip {1em} \newcommand\AIPparagraphafterindent {true} \newcommand\AIPparagraphrunin {true} \newcommand\AIPparagraphfont {\normalfont\normalsize\itshape} \newcommand\AIPparagraphformat[1] {#1.} \newcommand\AIPsubparagraphpenalty {-300} \newcommand\AIPsubparagraphindent {\bodytextparindent} \newcommand\AIPsubparagraphpreskip {\bodytextbaselineskip plus 3pt minus 1pt} \newcommand\AIPsubparagraphpostskip {1em} \newcommand\AIPsubparagraphafterindent {true} \newcommand\AIPsubparagraphrunin {true} \newcommand\AIPsubparagraphfont {\normalfont\normalsize} \newcommand\AIPsubparagraphformat[1] {#1.} \AIP@cmdnotsupported\subparagraph \setlength\leftmargini {2\bodytextparindent} \setlength\leftmarginii {2\bodytextparindent} \setlength\leftmarginiii {2\bodytextparindent} \setlength\leftmarginiv {2\bodytextparindent} \SetInternalRegister\@beginparpenalty {10000} \SetInternalRegister\@endparpenalty {-\@lowpenalty} \SetInternalRegister\@itempenalty {-\@lowpenalty} \renewcommand\@listi{ \setlength\leftmargin {\leftmargini} \setlength\labelsep {\bodytextenspace} \setlength\labelwidth {\leftmargin - \labelsep} \setlength\topsep {.5\bodytextbaselineskip plus 1pt} \setlength\partopsep {0pt} \setlength\parsep {0pt} \setlength\listparindent{0pt} \setlength\itemsep {2pt plus 1pt} } \renewcommand\@listii{ \setlength\leftmargin {\leftmarginii} \setlength\labelwidth {\leftmargin - \labelsep} \setlength\topsep {0pt plus 1pt} } \renewcommand\@listiii{ \setlength\leftmargin{\leftmarginiii} \setlength\labelwidth {\leftmargin - \labelsep} } \renewcommand\@listiv{ \setlength\leftmargin{\leftmarginiv} \setlength\labelwidth {\leftmargin - \labelsep} } \renewcommand\@listI{} \let\@listI\@listi \newcommand\@listIsmall{} \let\@listIsmall\@listi \newcommand\@listIfootnotesize{} \let\@listIsmall\@listi \renewcommand\labelitemi {\footnotesize\textbullet} \renewcommand\labelitemii {\bfseries --} \renewcommand\labelitemiii{\fontsize{8}{8}\selectfont \raisebox{1.5pt}\textasteriskcentered} \renewcommand\labelitemiv {\textperiodcentered} \renewcommand\labelenumi{\theenumi.} \renewcommand\labelenumii{(\theenumii)} \renewcommand\labelenumiii{\theenumiii.} \renewcommand{\labelenumiv}{\theenumiv.} \renewcommand\theenumi{\arabic{enumi}} \renewcommand\p@enumi {} \renewcommand\theenumii{\alph{enumii}} \renewcommand\p@enumii {\theenumi} \renewcommand\theenumiii{\roman{enumiii}} \renewcommand\p@enumiii {\theenumi(\theenumii)} \renewcommand{\theenumiv}{\Alph{enumiv}} \renewcommand{\p@enumiv} {\p@enumiii\theenumiii} \renewcommand*\descriptionlabel[1]{\hspace\labelsep \normalfont\bfseries #1} \newcommand\AIPeqformat [1] {(#1)} \newcommand\AIPeqfont {\normalfont \normalcolor} \newcommand\AIPeqrefformat [1] {(#1)} \newcommand\AIPeqreffont {} \newcommand\AIPfootnoterulewidth {2in} \newcommand\AIPfootnoteruleheight {0.5pt} \newcommand\AIPfootnoteruleindent {0pt} \newcommand\AIPfootnoterulepreskip {0.25in} \newcommand\AIPfootnoterulepostskip {0.125in} \renewcommand\thefootnote{\arabic{footnote}} \newcommand\AIPfootnotemarkerformat[1] {\textsuperscript{#1}} \newcommand\AIPfootnotemarkerfont {\fontsize{8}{8}\normalfont} \newcommand\AIPfootnotetextmarkerformat[1] {\textsuperscript{#1}\hspace{3pt}} \newcommand\AIPfootnotetextmarkerfont {\fontsize{8}{8}\normalfont} \setlength\footnotesep {8.4pt} \DeclareParagraphLayout{AIPfootnote} {10pt}{11pt} {1em}{0pt}{0pt}{0pt plus 1fil} {0pt} {\normalfont} \newcommand\AIPcitefont {} \newcommand\AIPciteformat[1] {[#1]} \setlength\arrayrulewidth {.4pt} \setlength\AIPhlinesep {1pt} \setlength{\skip\@mpfootins}{12pt plus 4pt minus 4pt} \setcounter{totalnumber} {3} \setcounter{topnumber} {2} \setcounter{bottomnumber}{1} \setcounter{dbltopnumber}{2} \renewcommand\topfraction {.9} \renewcommand\bottomfraction {.4} \renewcommand\textfraction {.08} \renewcommand\floatpagefraction{.9} \renewcommand\dbltopfraction {.6} \renewcommand\dblfloatpagefraction{.6} \setlength\floatsep {1.5\bodytextbaselineskip plus 4pt} \setlength\intextsep {.5\bodytextbaselineskip plus 2pt minus 1pt} \setlength\textfloatsep {2\bodytextbaselineskip plus 4pt} \setlength\@fptop {0pt plus 100pt} \setlength\@fpsep {20pt plus 6pt} \setlength\@fpbot {0pt plus 100pt} \setlength\@dblfptop {0pt plus 100pt} \setlength\@dblfpsep {20pt plus 6pt} \setlength\@dblfpbot {0pt plus 100pt} \renewcommand\figurename {FIGURE} \renewcommand{\thefigure} {\arabic{figure}} \newcommand\AIPfigurecaptionheadformat[1] {\figurename\ #1.\hspace{1em}} \setlength\AIPfigurecaptionBBskip {0.25in} \DeclareParagraphLayout{AIPfigure-singlelinecaption} {10pt}{11.5pt} {0pt}{\fill}{\fill}{0pt} {0pt} {} \DeclareParagraphLayout{AIPfigure-multilinecaption} {10pt}{11.5pt} {12pt}{0pt}{0pt}{0pt plus 1fil} {0pt} {\SetInternalRegister\hyphenpenalty{200} \SetInternalRegister\finalhyphendemerits{10000} } \newcommand\AIPfigurecaptionheadfont {\fontsize{10}{11.5}\bfseries} \newcommand\AIPfigurecaptiontextfont {\fontsize{10}{11.5}\selectfont} \newcommand\AIPfiguresourceheadtext {Source:~ } \newcommand\AIPfiguresourceskip {1mm} \newcommand\AIPfiguresourceheadfont {\fontsize{8}{8}\itshape} \newcommand\AIPfiguresourcetextfont {\fontsize{8}{8}\itshape} \newcommand\AIPtablefont {\fontsize{10}{11.5}\normalfont} \renewcommand\tablename {TABLE} \renewcommand{\thetable} {\arabic{table}} \newcommand\AIPtablecaptionheadformat[1] {\tablename\ #1.\hspace{1em}} \newcommand\AIPtablecaptionheadfont {\AIPtablefont\bfseries} \newcommand\AIPtablecaptiontextfont {\AIPtablefont} \setlength\AIPtablecaptionminwidth {12pc} \DeclareParagraphLayout{AIPtable-singlelinecaption} {10pt}{11.5pt} {0pt}{0pt}{0pt}{0pt plus 1fil} {0pt} {\normalfont} \setlength\AIPtablecaptionskip {2pt} \DeclareParagraphLayoutAlias{AIPtable-multilinecaption} {AIPtable-singlelinecaption} \newcommand\AIPtableheadfont {\AIPtablefont\bfseries} \setlength\AIPtablenoteskip {2mm} \DeclareParagraphLayout{AIPtablenote} {9pt}{10pt} {1em}{0pt}{0pt plus 3em}{0pt plus 1fil} {0pt} {\SetInternalRegister\hyphenpenalty{200} \SetInternalRegister\finalhyphendemerits{10000} } \newcommand\AIPtablesourceheadtext {Source:~ } % include spacing! \newcommand\AIPtablesourceskip {1mm} \newcommand\AIPtablesourceheadfont {\fontsize{8}{8}\itshape} \newcommand\AIPtablesourcetextfont {\fontsize{8}{8}\itshape} \newcommand\AIPfolioformat[3] {#1\hfil#2\hfil#3} \newcommand\AIPfoliofont {\fontsize{10}{10}\selectfont} \newcommand\AIPacknowledgmentsheadtype {\section*} \newcommand\AIPacknowledgmentsheadtext {ACKNOWLEDGMENTS} \DeclareParagraphLayoutAlias{AIPacknowledgments}{AIPbodytext} \newcommand\AIPindexheadtype {\section*} \renewcommand\indexname {INDEX} \newcommand\AIPbibliographyheadtype {\section*} \renewcommand\refname {REFERENCES} \renewcommand\AIPcitestyleselect {num} \newcommand\AIPnumcitestyle {aipproc} \newcommand\AIPbibliographymarkerformat [1] {\hfill#1.} \newcommand\AIPbibliographylabelsep {5pt} \newcommand\AIPbibliographyitemsep {0pt} \newcommand\AIPbibliographylabelwidth {\maxdimen} \newcommand\AIPbibliographyleftmargin {0pt} \newcommand\AIPbibliographyleftmarginextra {8pt} \DeclareParagraphLayout{AIPbibliography} {10pt}{11pt} {0pt}{0pt}{0pt}{0pt plus 1fil} {0pt} {\SetInternalRegister\hyphenpenalty{200} \SetInternalRegister\finalhyphendemerits{10000} \sloppy \frenchspacing \SetInternalRegister\clubpenalty {8000} \SetInternalRegister\widowpenalty{8000} } \xfm@ignored@key{homepage} \xfm@ignored@key{thanks} \xfm@ignored@key{email} \pagenumbering{arabic} \endinput %% %% End of file `aip-6s.clo'. ---------------0607241637508 Content-Type: application/x-tex; name="aipproc.cls" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aipproc.cls" %% %% This is file `aipproc.cls', %% generated with the docstrip utility. %% %% The original source files were: %% %% aipproc.dtx (with options: `class') %% %% Class aipproc to use with LaTeX2e %% (C) 1998,2000 American Institute of Physics and Frank Mittelbach %% All rights reserved %% %% Class aipproc to use with LaTeX2e %% %% Copyright (C) 1998, 2000, 2001, 2002, 2004, 2005 Frank Mittelbach %% Copyright (C) 1998, 2000, 2001, 2002, 2004, 2005 American Institute of Physics %% All rights reserved. %% %% Development of this class was commissioned by American Institute of Physics. %% \NeedsTeXFormat{LaTeX2e}[1999/06/01] \ProvidesClass{aipproc} [2005/11/11 v1.5a AIP Proceedings (FMi)] %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} %% \IfFileExists{fixltx2e.sty} {\RequirePackage{fixltx2e}} {\RequirePackage{fix2col}[1998/08/17]} \@ifpackageloaded{fixltx2e}{% \@ifpackagelater{fixltx2e}{1999/12/02}{}{% \def\addpenalty#1{% \ifvmode \if@minipage \else \if@nobreak \else \ifdim\lastskip=\z@ \penalty#1\relax \else \@tempskipb\lastskip \advance \@tempskipb \ifdim\prevdepth>\maxdepth\maxdepth\else \ifdim \prevdepth = -\@m\p@ \z@ \else \prevdepth \fi \fi \vskip -\@tempskipb \penalty#1% \vskip\@tempskipb \fi \fi \fi \else \@noitemerr \fi} \def \@doclearpage {% \ifvoid\footins \setbox\@tempboxa\vsplit\@cclv to\z@ \unvbox\@tempboxa \setbox\@tempboxa\box\@cclv \xdef\@deferlist{\@toplist\@botlist\@deferlist}% \global \let \@toplist \@empty \global \let \@botlist \@empty \global \@colroom \@colht \ifx \@currlist\@empty \else \@latexerr{Float(s) lost}\@ehb \global \let \@currlist \@empty \fi \@makefcolumn\@deferlist \@whilesw\if@fcolmade \fi{\@opcol\@makefcolumn\@deferlist}% \if@twocolumn \if@firstcolumn \xdef\@deferlist{\@dbltoplist\@deferlist}% \global \let \@dbltoplist \@empty \global \@colht \textheight \begingroup \@dblfloatplacement \@makefcolumn\@deferlist \@whilesw\if@fcolmade \fi{\@outputpage \@makefcolumn\@deferlist}% \endgroup \else \vbox{}\clearpage \fi \fi \ifx\@deferlist\@empty \else\clearpage \fi \else \setbox\@cclv\vbox{\box\@cclv\vfil}% \@makecol\@opcol \clearpage \fi } \def \@addtocurcol {% \@insertfalse \@setfloattypecounts \ifnum \@fpstype=8 \else \ifnum \@fpstype=24 \else \@flsettextmin \advance \@textmin \@textfloatsheight \@reqcolroom \@pageht \ifdim \@textmin>\@reqcolroom \@reqcolroom \@textmin \fi \advance \@reqcolroom \ht\@currbox \ifdim \@colroom>\@reqcolroom \@flsetnum \@colnum \ifnum \@colnum>\z@ \@bitor\@currtype\@deferlist \@testwrongwidth\@currbox \if@test \else \@bitor\@currtype\@botlist \if@test \@addtobot \else \ifodd \count\@currbox \advance \@reqcolroom \intextsep \ifdim \@colroom>\@reqcolroom \global \advance \@colnum \m@ne \global \advance \@textfloatsheight \ht\@currbox \global \advance \@textfloatsheight 2\intextsep \@cons \@midlist \@currbox \if@nobreak \nobreak \@nobreakfalse \everypar{}% \else \addpenalty \interlinepenalty \fi \vskip \intextsep \box\@currbox \penalty\interlinepenalty \vskip\intextsep \ifnum\outputpenalty <-\@Mii \vskip -\parskip\fi \outputpenalty \z@ \@inserttrue \fi \fi \if@insert \else \@addtotoporbot \fi \fi \fi \fi \fi \fi \fi \if@insert \else \@resethfps \@cons\@deferlist\@currbox \fi }}} {} \RequirePackage{calc} \RequirePackage{ifthen} \RequirePackage[final]{graphicx} \newif\if@load@natbib \@load@natbibtrue \IfFileExists{url.sty} {\RequirePackage{url}% } {\def\url##1{\texttt{##1}}% \ClassWarningNoLine{aipproc} {\noexpand\url command might fail with this LaTeX \MessageBreak installation since url.sty is missing}% } \IfFileExists{textcase.sty} {\RequirePackage{textcase}% } {\global\let\MakeTextUppercase\MakeUppercase \ClassWarningNoLine{aipproc} {\noexpand\section commands should not contain math as this on LaTeX \MessageBreak installation the textcase package is missing}% } \newcommand\AIP@optionnotsupported[1] {\ClassWarningNoLine{aipproc}% {Option~ `#1'~ not~ supported~ ---~ request~ ignored}} \newcommand\AIP@error{\ClassError{aipproc}} \newcommand\AIP@cmdnotsupported[1] {\def#1{\AIP@error{Command \noexpand#1not supported by class}\@eha}} \newcommand\AIP@natbibnotavailable[1] {\def#1{\AIP@error{Command \noexpand#1not supported if natbib not installed}\@eha}} \newcommand*\DesignerError[1]{% \AIP@error{#1}{Probably bug in class file.}} \newcommand*\InformationError[1]{% \AIP@error{#1}% {Add the necessary information to the document.}} \newcommand\MakeSpaceIgnore{% \catcode`\~=10\relax \catcode`\ = 9\relax \catcode`\^^M = 9\relax } \newcommand\MakeSpaceNormal{% \catcode`\~= 13\relax \catcode`\ = 10\relax \catcode`\^^M = 5\relax } \let\UnbreakableSpace~ \MakeSpaceIgnore \DeclareOption{a5paper} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{b5paper} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{legalpaper} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{executivepaper}{\AIP@optionnotsupported\CurrentOption} \DeclareOption{landscape} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{10pt} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{11pt} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{12pt} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{titlepage} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{notitlepage} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{oneside} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{twoside} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{onecolumn} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{twocolumn} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{leqno} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{fleqn} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{openbib} {\AIP@optionnotsupported\CurrentOption} \DeclareOption{tnotealph} {\def\AIP@tnote@representation{\@alph}} \DeclareOption{tnotesymbol}{\def\AIP@tnote@representation{\@fnsymbol}} \newboolean{@cmrfonts} \DeclareOption{cmfonts} {\setboolean{@cmrfonts}{true} \def\AIP@mathfontsused{0}} \DeclareOption{mathptm} {\def\AIP@mathfontsused{1}} \DeclareOption{mathtime} {\def\AIP@mathfontsused{2}} \DeclareOption{nomathfonts}{\def\AIP@mathfontsused{3}} \DeclareOption{mathptmx} {\def\AIP@mathfontsused{4}} \DeclareOption{mtpro} {\def\AIP@mathfontsused{5}} \def\pageref{0} \DeclareOption{varioref} {\def\pageref{1}} \DeclareOption{nonvarioref} {\def\pageref{2}} \DeclareOption{numcites} {\def\AIPcitestyleselect{num}} \DeclareOption{bibliocites} {\def\AIPcitestyleselect{biblio}} \DeclareOption{nonatbib} {\dont@load@natbibfalse} \DeclareOption{numberedheadings} {\AtEndOfClass{\setcounter{secnumdepth}{3}}} \DeclareOption{unnumberedheadings} {\AtEndOfClass{\setcounter{secnumdepth}{-\maxdimen}}} \DeclareOption{draft}{\PassOptionsToClass{\CurrentOption}{article}% \@drafttrue \AtEndOfPackage{ \let\AIP@pagenumerror\@gobble \def\@oddfoot{\reset@font \AIPfoliofont \AIPfolioformat\@shorttitle\@date\thepage }}} \newif\if@draft \DeclareOption{final}{\PassOptionsToClass{\CurrentOption}{article}} \DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} \ExecuteOptions{mathptmx,tnotesymbol,numcites,unnumberedheadings,letterpaper} \ProcessOptions\relax \MakeSpaceNormal \LoadClass{article} \MakeSpaceIgnore \def\layoutstyle#1{% \expandafter\let\expandafter \AIP@layoutstylename \csname AIP@layout@style@#1 \endcsname \ifx\AIP@layoutstylename\relax \def\AIP@layoutstylename{#1} \fi \MakeSpaceIgnore \makeatletter \InputIfFileExists{aip-\AIP@layoutstylename.clo} {\let\AIP@check@layoutstyle\relax} {\AIP@error{The~ layout~ style~ `#1'~ is~ not~ known\MessageBreak or~ its~ support~ file~ can~ not~ be~ found} {The~ \noexpand \layoutstyle command~ tried~ to~ load~ the~ file~ aip-\AIP@layoutstylename.clo~ without~ success!\MessageBreak This~ might~ be~ due~ to~ misspelling~ the~ style~ name.\MessageBreak Standard~ styles~ are~ `6x9',~ `8x11single',~ `8x11double',~ and~ `arlo',~ but\MessageBreak there~ might~ be~ others~ (see~ the~ class~ documentation).\MessageBreak It~ could~ also~ be~ due~ to~ an~ incomplete~ installation~ of~ the~ class. } } \MakeSpaceNormal \makeatother \ifdim\columnsep>\z@ \@twocolumntrue \else \@twocolumnfalse \fi } \@onlypreamble\layoutstyle \def\declare@layoutstyle#1#2{ \@namedef{AIP@layout@style@#1}{#2} } \@onlypreamble\declare@layoutstyle \declare@layoutstyle{6x9}{6s} \declare@layoutstyle{8x11single}{8s} \declare@layoutstyle{8x11double}{8d} \def\AIP@check@layoutstyle{ \AIP@error{No~ \noexpand\layoutstyle command~ seen} {The~ class~ requires~ a~ \noexpand\layoutstyle{}~ declaration~ in~ the~ preamble!\MessageBreak Standard~ styles~ are~ `6x9',~ `8x11single',~ `8x11double',~ and~ `arlo',~ but\MessageBreak there~ might~ be~ others~ (see~ the~ class~ documentation).\MessageBreak To~ be~ able~ to~ proceed~ the~ 6x9~ style~ is~ assumed. } \layoutstyle{6x9} \@colht\textheight \@colroom\textheight \vsize\textheight \columnwidth\textwidth \@clubpenalty\clubpenalty \if@twocolumn \advance\columnwidth -\columnsep \divide\columnwidth\tw@ \hsize\columnwidth \@firstcolumntrue \fi \hsize\columnwidth \linewidth\hsize } \AtBeginDocument{\AIP@check@layoutstyle} \newcommand*\SetInternalRegister[2]{#1=#2\relax} \let\SetInternalCounter\count@assign \newcommand*\DeclareParagraphLayout[9]{% \@namedef{#1Para}{ \fontsize{#2}{#3}\selectfont #9 \setlength\parindent {#4} \setlength\leftskip {#5} \setlength\rightskip {#6} \@rightskip\rightskip \setlength\parfillskip{#7} \setlength\parskip {#8} } } \@onlypreamble\DeclareParagraphLayout \newcommand*\UseParagraphLayout[1]{ \@ifundefined{#1Para} {\DesignerError{Paragraph~ layout~ '#1'~ undefined}} {\@nameuse{#1Para}} } \newcommand*\DeclareParagraphLayoutAlias[2]{% \@ifundefined{#2Para} {\DesignerError{Paragraph~ layout~ '#2'~ undefined}} {\expandafter\let \csname#1Para\expandafter\endcsname \csname#2Para\endcsname } } \@onlypreamble\DeclareParagraphLayoutAlias \newcommand*\UseBBskip[1] {\ifvmode \setlength\@tempskipa{#1 - \parskip - \baselineskip} \vskip\@tempskipa \else \DesignerError{\protect\UseBBskip\space outside~ vmode} \fi } \newcommand*\DeclarePagestyle[5] { \@namedef{ps@#1} { \def\@oddhead {#2} \def\@oddfoot {#3} \def\@evenhead{#4} \def\@evenfoot{#5} } } \newdimen\bodytextsize \newdimen\bodytextbaselineskip \newdimen\bodytextenspace \newdimen\bodytextparindent \pagestyle{empty} \AIP@cmdnotsupported\pagestyle \newcommand\AIP@pagenumerror[1]{% \AIP@error{Command~ \string#1~ can't~ be~ used~ in~ production}% {This~ command~ will~ produce~ page~ numbers~ which~ will~ be~ incorrect~ in~ the\MessageBreak final~ production. It~ should~ therefore~ only~ be~ used~ while~ producing~ drafts.}} \let\@@tableofcontents\tableofcontents \let\@@listoffigures\listoffigures \let\@@listoftables\listoftables \renewcommand\tableofcontents{% \AIP@pagenumerror\tableofcontents\@@tableofcontents} \renewcommand\listoffigures{% \AIP@pagenumerror\listoffigures\@@listoffigures} \renewcommand\listoftables{% \AIP@pagenumerror\listoftables\@@listoftables} \RequirePackage{aipxfm} \MakeSpaceIgnore \def\AIP@startsection#1#2#3#4#5{ \@tempskipa#2\relax \advance\@tempskipa-\parskip \ifdim\@tempskipa<\z@ \DesignerError{#2~ -~ \protect\parskip needs~ to~ be~ non-negative} \fi \ifthenelse{\equal#1{true}} \relax {\@tempskipa-\@tempskipa} \edef\AIP@preskip{\the\@tempskipa} \@tempskipa#4\relax \advance\@tempskipa-\parskip \ifdim\@tempskipa<\z@ \DesignerError{#2~ -~ \protect\parskip needs~ to~ be~ non-negative} \fi \ifthenelse{\equal#3{true}} {\@tempskipa-\@tempskipa} \relax \edef\AIP@postskip{\the\@tempskipa} \@secpenalty#5\relax \@startsection } \renewcommand\section {\AIP@startsection \AIPsectionafterindent\AIPsectionpreskip \AIPsectionrunin\AIPsectionpostskip \AIPsectionpenalty {section}{1}{\AIPsectionindent}% {\AIP@preskip}% {\AIP@postskip}% {\AIPsectionfont\AIPsectionformat}} \renewcommand\subsection {\AIP@startsection \AIPsubsectionafterindent\AIPsubsectionpreskip \AIPsubsectionrunin\AIPsubsectionpostskip \AIPsubsectionpenalty {subsection}{2}{\AIPsubsectionindent}% {\AIP@preskip}% {\AIP@postskip}% {\AIPsubsectionfont\AIPsubsectionformat}} \renewcommand\subsubsection {\AIP@startsection \AIPsubsubsectionafterindent\AIPsubsubsectionpreskip \AIPsubsubsectionrunin\AIPsubsubsectionpostskip \AIPsubsubsectionpenalty {subsubsection}{3}{\AIPsubsubsectionindent}% {\AIP@preskip}% {\AIP@postskip}% {\AIPsubsubsectionfont\AIPsubsubsectionformat}} \renewcommand\paragraph {\AIP@startsection \AIPparagraphafterindent\AIPparagraphpreskip \AIPparagraphrunin\AIPparagraphpostskip \AIPparagraphpenalty {paragraph}{4}{\AIPparagraphindent}% {\AIP@preskip}% {\AIP@postskip}% {\AIPparagraphfont\AIPparagraphformat}} \renewcommand\subparagraph {\AIP@startsection \AIPsubparagraphafterindent\AIPsubparagraphpreskip \AIPsubparagraphrunin\AIPsubparagraphpostskip \AIPsubparagraphpenalty {subparagraph}{5}{\AIPsubparagraphindent}% {\AIP@preskip}% {\AIP@postskip}% {\AIPsubparagraphfont\AIPsubparagraphformat}} \newcommand\UseNoHyphens{\hyphenpenalty\@M\exhyphenpenalty\@M} \ifcase \AIP@mathfontsused % 0 use cm for everything \or \MakeSpaceNormal \RequirePackage{mathptm} % 1 \MakeSpaceIgnore \or \MakeSpaceNormal \RequirePackage{mathtime} % 2 \MakeSpaceIgnore \or % 3 use cm for math \or \MakeSpaceNormal \RequirePackage{mathptmx} % 4 \MakeSpaceIgnore \or \MakeSpaceNormal \RequirePackage{mtpro} % 5 \MakeSpaceIgnore \fi \ifnum \AIP@mathfontsused > 0 \RequirePackage{times} \normalfont \RequirePackage[T1]{fontenc} \RequirePackage{textcomp} \fi \AtBeginDocument{\UseParagraphLayout{AIPbodytext}} \renewcommand\footnoterule{ \setlength\skip@{\AIPfootnoteruleheight+\AIPfootnoterulepostskip} \vskip-\skip@ \moveright \AIPfootnoteruleindent\vbox{% \hrule \@width \AIPfootnoterulewidth \@height \AIPfootnoteruleheight}% \vskip \AIPfootnoterulepostskip \relax} \AtBeginDocument{ \setlength{\skip\footins}{\AIPfootnoterulepreskip +\AIPfootnoterulepostskip}} \renewcommand\@makefntext[1]{ \UseParagraphLayout{AIPfootnote} \noindent \hbox{\AIPfootnotetextmarkerformat {\AIPfootnotetextmarkerfont\@thefnmark}}% \ignorespaces #1} \def\@makefnmark{\hbox{% \AIPfootnotemarkerformat{\AIPfootnotemarkerfont\@thefnmark}}} \def \@makecol {% \setbox\@outputbox \box\@cclv \@combinefloats \ifvoid\footins \else \setbox\@outputbox \vbox {% \boxmaxdepth \@maxdepth \unvbox \@outputbox \vskip \skip\footins \color@begingroup \normalcolor \footnoterule \unvbox \footins \color@endgroup }% \fi \xdef\@freelist{\@freelist\@midlist}% \global \let \@midlist \@empty \ifvbox\@kludgeins \@makespecialcolbox \else \setbox\@outputbox \vbox to\@colht {% \@texttop \dimen@ \dp\@outputbox \unvbox \@outputbox \vskip -\dimen@ \@textbottom }% \fi \global \maxdepth \@maxdepth } \def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or **\or \ddagger\or \mathsection\or \mathparagraph\or \|\or \dagger\dagger \or \ddagger\ddagger \or\mathsection\mathsection \or \mathparagraph\mathparagraph \or *{*}*\or \dagger{\dagger}\dagger \or\ddagger{\ddagger}\ddagger\or \mathsection{\mathsection}\mathsection \or \mathparagraph{\mathparagraph}\mathparagraph \else\@ctrerr\fi}} \def\@alph#1{\ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or y\or z\or aa\or bb\or cc\or dd\or ee\or ff\or gg\or hh\or ii\or jj\or kk\or ll\or mm\or nn\or oo\or pp\or qq\or rr\or ss\or tt\or uu\or vv\or ww\or xx\or yy\or zz\else\@ctrerr\fi} \AtBeginDocument{% \ifx\tagform@\@undefined \def\eqref#1{\mbox{\AIPeqreffont\AIPeqrefformat{\ref{#1}}}}% \else \def\tagform@#1{\mbox{\AIPeqreffont \AIPeqrefformat{\ignorespaces #1\unskip\@@italiccorr}}}% \fi \def\@eqnnum{{\AIPeqfont\AIPeqformat\theequation}} } \ifnum\pageref>0 \MakeSpaceNormal \RequirePackage{varioref} \MakeSpaceIgnore \renewcommand\reftextfaceafter {on~ the~ next~ page} \renewcommand\reftextfacebefore{on~ the~ \reftextvario{previous} {preceding}~ page} \renewcommand\reftextafter {on~ the~ \reftextvario{following} {next}~ page} \renewcommand\reftextbefore {on~ the~ \reftextvario{preceding~ page} {page~ before}} \renewcommand\reftextcurrent {on~ \reftextvario{this}% {the~ current}~ page} \renewcommand\reftextfaraway[1]{% \is@pos@number\@tempb {\ifnum\@tempb<0\@tempa\relax \reftextearlier \else \reftextlater \fi}% {\@setref\relax\relax{#1}}} \newcommand\reftextearlier{\reftextvario{on~ an~ earlier~ page} {earlier~ on}} \newcommand\reftextlater {\reftextvario{later~ on}{further~ down}} \ifnum\pageref=2 \def\reftextvario#1#2{#1} \fi \let\pageref\vpageref \else \renewcommand\pageref[1] {\AIP@error{Page~ references~ not~ supported} {This~ class~ does~ not~ support~ references~ to~ page~ numbers~ unless~ the~ varioref~ or~ the~ nonvarioref~ option~ is~ used,~ since~ it~ doesn't~ print~ page~ numbers.}} \fi \newcommand\AIP@maketablecaption[2]{% \UseParagraphLayout{AIPtable-singlelinecaption} \settowidth\@tempdima{% \noindent {\AIPtablecaptionheadfont\AIPtablecaptionheadformat{#1}} \AIPtablecaptiontextfont\ignorespaces#2} \ifdim\@tempdima>\hsize \UseParagraphLayout{AIPtable-multilinecaption} \fi \noindent {\AIPtablecaptionheadfont\AIPtablecaptionheadformat{#1}} \AIPtablecaptiontextfont\ignorespaces#2\par \vskip\AIPtablecaptionskip} \newskip\AIPtablecaptionskip \newcommand\AIP@makefigurecaption[2]{% \UseParagraphLayout{AIPfigure-singlelinecaption} \UseBBskip\AIPfigurecaptionBBskip \settowidth\@tempdima{% \noindent {\AIPfigurecaptionheadfont\AIPfigurecaptionheadformat{#1}} \AIPfigurecaptiontextfont\ignorespaces#2} \ifdim\@tempdima>\hsize \UseParagraphLayout{AIPfigure-multilinecaption} \fi \noindent {\AIPfigurecaptionheadfont\AIPfigurecaptionheadformat{#1}} \AIPfigurecaptiontextfont\ignorespaces#2\par } \newskip\AIPfigurecaptionBBskip \newcommand\AIP@sourceerror{\AIP@error {\noexpand\source is only supported with `table' or `figure' environment}\@ehd} \let\source\AIP@sourceerror \newcommand\AIP@fsource@setup{% \def\source##1{\gdef\AIP@typeset@source {\addvspace\AIPfiguresourceskip \rightline{\AIPfiguresourceheadfont \AIPfiguresourceheadtext \AIPfiguresourcetextfont ##1} }} \global\let\AIP@typeset@source\@empty} \newcommand\AIP@tsource@setup{% \def\source##1{\gdef\AIP@typeset@source {\addvspace\AIPtablesourceskip \rightline{\AIPtablesourceheadfont \AIPtablesourceheadtext \AIPtablesourcetextfont ##1} }} \global\let\AIP@typeset@source\@empty} \newcommand\AIP@tablenoteerror{\AIP@error {\noexpand\tablenote is only supported inside `table' environment\MessageBreak and not allowed inside the \noexpand\caption or \noexpand\source command}\@ehd} \let\tablenote\AIP@tablenoteerror \newcommand\AIP@tablenote[2]{% \leavevmode \stepcounter\@mpfn \protected@xdef\@thefnmark{\thempfn}% #1\@footnotemark \protected@xdef\AIP@tnote@process {\AIP@tnote@process \protect\footnotetext [\the\c@mpfootnote] {\protect\UseParagraphLayout{AIPtablenote}#2}}% } \newcommand\AIP@tnote@setup{% \def\@mpfn{mpfootnote}% \def\thempfn{\thempfootnote}% \def\thempfootnote{\AIP@tnote@representation\c@mpfootnote}% \global\c@mpfootnote\z@ \def\tablenote{\@ifstar{\AIP@tablenote\relax} {\AIP@tablenote\rlap}} \gdef\AIP@tnote@process{}% \setlength{\skip\@mpfootins}{\AIPtablenoteskip} \let\footnoterule\relax \let\@footnotetext\@mpfootnotetext } \newskip\AIPtablenoteskip \newcommand\AIP@tablehead[4]{\multicolumn{#1}{#2}% {\AIPtableheadfont\begin{tabular}[#3]{@{}#2@{}}% \vrule \@height \bodytextsize\@width \z@\relax \ignorespaces#4\unskip \vrule \@depth .5\bodytextsize\@width \z@\end{tabular}}} \def\hline{% \noalign{\ifnum0=`}\fi\vskip\AIPhlinesep \hrule \@height \arrayrulewidth\vskip3\AIPhlinesep \futurelet \reserved@a\@xhline} \newdimen\AIPhlinesep \newenvironment{ltxtable}[1][tbp] {\@float{table}[#1] \let\tablehead\AIP@tablehead \let\@makecaption\AIP@maketablecaption \AIPtablefont} {\end@float} \newenvironment{ltxtable*}[1][tbp] {\@dblfloat{table}[#1] \let\tablehead\AIP@tablehead \let\@makecaption\AIP@maketablecaption \AIPtablefont} {\end@dblfloat} \renewenvironment{table*}[1][tbp] {\AIP@error{Environment `table*' not supported\MessageBreak --- environment `table' used instead}% {The class automatically determines the position of the float according\MessageBreak to its size.}% \begin{table}} {\end{table}} \renewenvironment{table}[1][tbp] {\def\AIP@floatspec{#1}% \let\tablehead\AIP@tablehead \let\@makecaption\AIP@maketablecaption \AIP@tsource@setup \AIP@tnote@setup \global \setbox\AIP@box \color@hbox \hbox \bgroup \@floatboxreset \def\caption##1{\gdef\AIP@save@caption{##1}\let\caption\AIP@caption@error}% \def\label##1{\gdef\AIP@save@label{##1}}% \global\let\AIP@save@caption\@undefined \global\let\AIP@save@label\@undefined \normalcolor \AIPtablefont \ignorespaces } {% \AIP@remove@any@previous@space \outer@nobreak \egroup \color@endbox \setlength\dimen@{\columnwidth+1pt}% \ifdim\wd\AIP@box >\dimen@ \setlength\dimen@{\textwidth+1pt}% \ifdim\wd\AIP@box >\dimen@ \def\@captype{table}% \sbox\@tempboxa{\AIP@make@table@body}% \setlength\dimen@{\ht\@tempboxa+\dp\@tempboxa}% \ifdim\dimen@ <\columnwidth \def\@tempa{\@float{table}}% \expandafter\@tempa\expandafter[\AIP@floatspec]% \centerline{\rotatebox{90}{\box\@tempboxa}}% \end@float \else \def\@tempa{\@dblfloat{table}}% \expandafter\@tempa\expandafter[\AIP@floatspec]% \setbox\@tempboxa\hbox{\rotatebox{90}{\box\@tempboxa}} \dimen@\wd\@tempboxa \advance\dimen@ -5\p@ % grace \ifdim \dimen@ >\textwidth \AIP@error{Table~ too~ wide~ (\the\wd\@tempboxa\space >~\the\textwidth)} {Table~ doesn't~ fit~ even~ after~ turning~ it~ by~ 90~ degrees.~ You~ probably\MessageBreak have~ to~ change~ it~ somewhat.} \fi \centerline{\unhbox \@tempboxa} \end@dblfloat \fi \else \def\@tempa{\@dblfloat{table}}% \expandafter\@tempa\expandafter[\AIP@floatspec]% \AIP@make@table@body \end@dblfloat \fi \else \def\@tempa{\@float{table}}% \expandafter\@tempa\expandafter[\AIP@floatspec]% \AIP@make@table@body \end@float \fi } \newcommand\AIP@caption@error{\AIP@error{Only~ one~ \noexpand\caption command~ per~ float~ supported} {If~ you~ need~ more~ than~ one~ \noexpand\caption~ command~ try~ the~ ltxfigure~ or~ ltxtable\MessageBreak environment~ as~ explained~ in~ the~ aipguide.}} \newcommand\AIP@make@table@body{% \centering \@tempdima\wd\AIP@box \ifdim\@tempdima<\AIPtablecaptionminwidth \@tempdima\AIPtablecaptionminwidth \fi \begin{minipage}\@tempdima \ifx\AIP@save@caption\@undefined\else \let\tablenote\AIP@tablenoteerror \caption{\AIP@save@caption \ifx\AIP@save@label\@undefined\else \label\AIP@save@label \fi}% \fi \par \offinterlineskip % or we get \lineskip \vbox{\hsize\wd\AIP@box \box\AIP@box \AIP@typeset@source}% \AIP@tnote@process \end{minipage}% } \newdimen\AIPtablecaptionminwidth \newcommand\AIP@remove@any@previous@space {\unskip\loop\unskip\ifdim\lastskip>\z@\repeat} \newbox\AIP@box \newenvironment{ltxfigure}[1][tbp] {\@float{figure}[#1] \let\@makecaption\AIP@makefigurecaption} {\end@float} \newenvironment{ltxfigure*}[1][tbp] {\@dblfloat{figure}[#1] \let\@makecaption\AIP@makefigurecaption} {\end@dblfloat} \renewenvironment{figure}[1][tbp] {\def\AIP@floatspec{#1}% \AIP@fsource@setup \global \setbox\AIP@box \color@hbox \hbox \bgroup \@floatboxreset \def\caption##1{\let\caption\AIP@caption@error\gdef\AIP@save@caption{##1}}% \def\label##1{\gdef\AIP@save@label{##1}}% \global\let\AIP@save@caption\@undefined \global\let\AIP@save@label\@undefined \normalcolor \normalfont \normalsize \ignorespaces } {% \AIP@remove@any@previous@space \outer@nobreak \egroup \color@endbox \def\@tempa{\@dblfloat{figure}}% \setlength\dimen@{\columnwidth+1pt}% \ifdim\wd\AIP@box >\dimen@ \expandafter\@tempa\expandafter[\AIP@floatspec]% \AIP@make@figure@body \end@dblfloat \else \def\@tempa{\@float{figure}}% \expandafter\@tempa\expandafter[\AIP@floatspec]% \AIP@make@figure@body \end@float \fi } \renewenvironment{figure*}{\figure}{\endfigure} \newcommand\AIP@make@figure@body{% \centering \setlength\@tempdima{\wd\AIP@box-1pt}% \ifdim\@tempdima>\columnwidth \@tempdima\textwidth \else \@tempdima\columnwidth \fi \begin{minipage}\@tempdima \centerline{\vbox{\hsize\wd\AIP@box \box\AIP@box \AIP@typeset@source}}% \ifx\AIP@save@caption\@undefined\else \let\@makecaption\AIP@makefigurecaption \caption{\AIP@save@caption \ifx\AIP@save@label\@undefined\else \label\AIP@save@label \fi}% \fi \par \end{minipage}} \renewcommand\fnum@figure{\thefigure} \renewcommand\fnum@table{\thetable} \newcommand\spaceforfigure[2]{\parbox{#1}{\mbox{}\vspace*{#2}}} \AtBeginDocument{% \newcommand\@@longtable{}% \let\@@longtable\longtable \def\longtable{% \begingroup \LTcapwidth\z@ \advance\c@LT@tables\@ne % local as stepcounter comes later \let\LT@entry\AIP@get@longtable@width \csname LT@\romannumeral\c@LT@tables\endcsname \ifdim \LTcapwidth=\z@ \global \LTcapwidth2in \else \global\LTcapwidth\LTcapwidth \fi \endgroup \AIPtablefont \let\tablehead\AIP@tablehead \@@longtable }% \def\LT@makecaption#1#2#3{% \LT@mcol\LT@cols {@{}l@{}}{\rlap{\parbox[t]\LTcapwidth{% \UseParagraphLayout{AIPtable-multilinecaption} \noindent {\AIPtablecaptionheadfont\AIPtablecaptionheadformat{#1#2}} \AIPtablecaptiontextfont\ignorespaces#3\endgraf \vspace*\AIPtablecaptionskip }}}}% \def\LT@hline{% \noalign{\ifnum0=`}\fi \penalty\@M\vskip\AIPhlinesep \futurelet\@let@token\LT@@hline} \def\LT@@hline{% \ifx\@let@token\hline \global\let\@gtempa\@gobble \gdef\LT@sep{\penalty-\@medpenalty\vskip\doublerulesep}% \else \global\let\@gtempa\@empty \gdef\LT@sep{\penalty-\@lowpenalty\vskip-\arrayrulewidth}% \fi \ifnum0=`{\fi}% \multispan\LT@cols \unskip\leaders\hrule\@height\arrayrulewidth\hfill\cr \noalign{\LT@sep}% \multispan\LT@cols \unskip\leaders\hrule\@height\arrayrulewidth\hfill\cr \noalign{\penalty\@M\vskip3\AIPhlinesep}% \@gtempa} } \newcommand\AIP@get@longtable@width[2]{\advance\LTcapwidth#2\relax } \newenvironment{theacknowledgments} {\AIPacknowledgmentsheadtype\AIPacknowledgmentsheadtext \UseParagraphLayout{AIPacknowledgments}} {\par} \renewenvironment{theindex} { \if@twocolumn \@restonecolfalse \AIPindexheadtype\indexname \else \columnseprule \z@ \columnsep 35\p@ \@restonecoltrue \twocolumn[\AIPindexheadtype\indexname] \fi \parindent\z@ \parskip\z@ \@plus .3\p@\relax \let\item\@idxitem} {\if@restonecol\onecolumn\fi} \IfFileExists{natbib.sty} { \if@load@natbib \AtBeginDocument{ \citestyle {\csname AIP \AIPcitestyleselect citestyle\endcsname} } \MakeSpaceNormal \RequirePackage{natbib} \MakeSpaceIgnore \fi }{} \ifx\citet\@undefined \def\citet{\AIP@natbibnotavailable\citet} \def\citep{\AIP@natbibnotavailable\citep} \def\citealt{\AIP@natbibnotavailable\citealt} \def\citealp{\AIP@natbibnotavailable\citealp} \def\citetext{\AIP@natbibnotavailable\citetext} \def\citeauthor{\AIP@natbibnotavailable\citeauthor} \def\citeyear{\AIP@natbibnotavailable\citeyear} \def\citeyearpar{\AIP@natbibnotavailable\citeyearpar} \def\Citet{\AIP@natbibnotavailable\Citet} \def\Citep{\AIP@natbibnotavailable\Citep} \def\Citealt{\AIP@natbibnotavailable\Citealt} \def\Citealp{\AIP@natbibnotavailable\Citealp} \def\Citetext{\AIP@natbibnotavailable\Citetext} \def\Citeauthor{\AIP@natbibnotavailable\Citeauthor} \def\Citeyear{\AIP@natbibnotavailable\Citeyear} \def\Citeyearpar{\AIP@natbibnotavailable\Citeyearpar} \renewenvironment{thebibliography}[1] {\AIPbibliographyheadtype\refname \list{\AIPbibliographymarkerformat{\@arabic\c@enumiv}}% {\settowidth\labelwidth{\AIPbibliographymarkerformat{#1}}% \UseParagraphLayout{AIPbibliography} \leftmargin\labelwidth \advance\leftmargin\labelsep \setlength\itemsep\AIPbibliographyitemsep \@openbib@code \usecounter{enumiv}% \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}% \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty `thebibliography' environment}}% \endlist} \renewcommand*\@cite[2]{{\AIPcitefont \AIPciteformat{#1\if@tempswa , #2\fi}}} \AtBeginDocument{ \ifthenelse{\equal\AIPcitestyleselect{num}} {} {\AIP@error{Author/year~ citation~ style~ impossible} {Without~ the~ natbib~ system~ only~ numerical~ citations~ are~ possible.~ But~ you~ (or~ the~ journal~ \string\layoutstyle)~ requested~ a~ diffferent~ citation~ style.} } } \else \renewcommand\bibsection{\AIPbibliographyheadtype\refname} \renewcommand\NAT@bibsetnum[1]{ \ifdim\AIPbibliographylabelwidth=\maxdimen \settowidth\labelwidth{\@biblabel{#1}} \setlength\leftmargin\labelwidth \else \setlength\labelwidth\AIPbibliographylabelwidth \setlength\leftmargin {\AIPbibliographyleftmargin+\AIPbibliographyleftmarginextra} 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\def\XFMauthorsbyaddress#1{ \let\XFMaddressmark\relax \edef\xfm@author@count{\the\xfm@authors} \def\@elt##1{ \let\xfm@address##1 #1 } \count@\z@ \xfm@addresslist } \def\XFMaddresses{ \let\XFMaddressmark\relax \par \begingroup \UseParagraphLayout{XFMaddress} \UseBBskip\XFMaddressBBskip \edef\xfm@author@count{\the\xfm@authors} \let\@elt\xfm@address@elt \count@\z@ \xfm@addresslist \par \endgroup } \def\xfm@address@elt#1{ \advance\count@\@ne \ifnum\xfm@authors>\@ne % no mark for a single author \ifnum\xfm@addresses>\@ne % or a single address for all authors \begingroup \c@xfm\count@ \XFMaddressmarkformat{ \XFMaddressmarkfont \XFMaddressmarkstyle{xfm} } \endgroup \fi \fi #1\XFMaddressseparator} \def\XFMauthorsoneaddress{ \def\xfm@address@test\ifx{ \expandafter\ifx\csname address-\XFMthe{addressnum}\endcsname\xfm@address} \xfm@address@loop} \def\XFMauthors{ \edef\xfm@author@count{\the\xfm@authors} \def\xfm@address@test\ifx{\iftrue} \xfm@address@loop} \def\xfm@address@loop{ \par \begingroup 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\newlength\XFMabstractleftindent \newskip\XFMtitleBBskip \def\XFMtitle{ \begingroup \UseParagraphLayout{XFMtitle} \UseBBskip\XFMtitleBBskip \XFMtitleformat{\ignorespaces\@title \par} \endgroup } \newskip\XFMsubtitleBBskip \def\XFMsubtitle{ \ifx\@subtitle\@empty \else \begingroup \UseParagraphLayout{XFMsubtitle} \UseBBskip\XFMsubtitleBBskip \XFMsubtitleformat{\ignorespaces\@subtitle \par} \endgroup \fi } \newcommand\XFMsubtitleformat[1] {#1} \DeclareParagraphLayout{XFMsubtitle} {14pt}{16pt} {0pt}{\fill}{\fill}{0pt} {0pt} {\SetInternalRegister\hyphenpenalty{2000} \SetInternalRegister\finalhyphendemerits{20000} \normalfont } \setlength\XFMsubtitleBBskip {14pt} \def\XFMcopyright{ \par \begingroup \UseParagraphLayout{XFMcopyrighttext} \UseBBskip\XFMcopyrightBBskip \noindent \XFMcopyrightformat\xfm@copyrightyear\xfm@copyrightholder \par \endgroup} \newskip\XFMcopyrightBBskip \def\copyrightholder#1{\def\xfm@copyrightholder{#1}} \def\copyrightyear#1{\def\xfm@copyrightyear{#1}} 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\expandafter\XFMdateheading\expandafter{\xfm@temp} \csname xfm@\xfm@temp\endcsname \fi} \par \endgroup} \newskip\XFMdatesBBskip \def\XFMdateheading#1{ \noindent\textbf{Date~\MakeUppercase #1:~}} \def\XFMdatesep{\hspace{1em}} \DeclareParagraphLayout{XFMdates} {9pt}{10pt} {0pt}{\bodytextparindent}{\bodytextparindent}{0pt plus 1fil} {0pt} {\normalfont} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \def\XFMdates#1{ \par \begingroup \UseParagraphLayout{XFMdatestext} \UseBBskip\XFMdatesBBskip \let\xfm@tempb\relax \@for\xfm@temp:=#1\do{ \expandafter\ifx\csname xfm@\xfm@temp\endcsname\relax \else \xfm@tempb \let\xfm@tempb\XFMdatesep \expandafter\XFMdateheading\expandafter{\xfm@temp} \csname xfm@\xfm@temp\endcsname \fi} \par \endgroup} \MakeSpaceNormal \endinput %% %% End of file `aipxfm.sty'. ---------------0607241637508 Content-Type: application/x-tex; name="cinvestav.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cinvestav.tex" \documentclass[draft]{aipproc} \usepackage{amsfonts} \usepackage{epsf} %& use ,final for the camera ready runs %% ,draft % use draft while you are working on the paper %% ,numberedheadings % uncomment this option for numbered sections %% , % add further options here if necessary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARROWS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\llra{\relbar\joinrel\longrightarrow} %THIS IS LONG \def\mapright#1{\smash{\mathop{\llra}\limits_{#1}}} %ARROW ON LINE \def\mapup#1{\smash{\mathop{\llra}\limits^{#1}}} %CAN PUT SOMETHING OVER IT \def\mapupdown#1#2{\smash{\mathop{\llra}\limits^{#1}_{#2}}} %over&under it% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END ARROWS %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% These are the AMS constructs for multiline limits %%%%%%%%%%%%% % \catcode`\@=11 \def\BF#1{{\bf {#1}}} \def\NEG#1{{\rlap/#1}} \def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi} \def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}} \def\multilimits@{\bgroup\vspace@\Let@ \baselineskip\fontdimen10 \scriptfont\tw@ \advance\baselineskip\fontdimen12 \scriptfont\tw@ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ \lineskiplimit\lineskip \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr} \def\Sb{_\multilimits@} \def\endSb{\crcr\egroup\egroup\egroup} \def\Sp{^\multilimits@} \let\endSp\endSb % %%%%%%%%%%%%%%%%%%%%END of explanations for multiline limits %%%%%%%%%%%%%%%%% \layoutstyle{6x9} \begin{document} \title[]{Description of resonances within the rigged Hilbert space} \classification{03.65.-w, 02.30.Hq} \keywords{Resonances; Hilbert space; rigged Hilbert space; Gamow states} \author{Rafael de la Madrid}{ address={Department of Physics, University of California at San Diego, La Jolla, CA 92093}, altaddress={E-mail: \texttt{rafa@physics.ucsd.edu}} } \begin{abstract} The spectrum of a quantum system has in general bound, scattering and resonant parts. The Hilbert space includes only the bound and scattering spectra, and discards the resonances. One must therefore enlarge the Hilbert space to a rigged Hilbert space, within which the physical bound, scattering and resonance spectra are included on the same footing. In these lectures, I will explain how this is done. \end{abstract} \date{} \maketitle \section{Introduction: Lecture~1} \label{sec:intro} In Quantum Mechanics, observable quantities are represented by linear operators. The eigenvalues of an operator represent the possible values of the measurement of the corresponding observable. These eigenvalues, which mathematically correspond to the spectrum of the operator, can be discrete (as the energies of a particle in a box), continuous (as the energies of a free, unconstrained particle), resonant (as in $\alpha$ decay), or a combination thereof. The Hilbert space includes only the bound and scattering spectra, because the Hilbert space spectrum of an observable is real, thereby discarding the resonance spectrum as unphysical. However, radioactive nuclei and unstable elementary particles are physical objects that ought to have a place in the quantum mechanical formalism. This is why we need to extend the Hilbert space to a rigged Hilbert space, within which the resonance spectrum has a place. The purpose of this series of lectures is to explain how one should use the rigged Hilbert space in quantum mechanics and, in particular, how to incorporate the resonance spectrum into the quantum mechanical formalism by using the rigged Hilbert space. When the spectrum of an observable $A$ is discrete and $A$ is bounded, then $A$ is defined on the whole of the Hilbert space $\cal H$ and the eigenvectors of $A$ belong to $\cal H$. In this case, $A$ can be essentially seen as a matrix. This means that, as far as discrete spectrum is concerned, there is no need to extend $\cal H$. However, quantum mechanical observables are in general unbounded and their spectrum has in general a continuous part. In order to deal with continuous spectrum, we use Dirac's bra-ket formalism. This formalism does not fit within the Hilbert space alone, but within the rigged Hilbert space. Loosely speaking, a rigged Hilbert space (also called a Gelfand triplet) is a triad of spaces \begin{equation} {\Phi} \subset {\cal H} \subset {\Phi}^{\times} \label{RHStIntro} \end{equation} such that $\cal H$ is a Hilbert space, $\Phi$ is a dense subspace of $\cal H$, and $\Phi ^{\times}$ is the space of antilinear functionals over $\Phi$. Mathematically, $\Phi$ is the space of test functions, and $\Phi ^{\times}$ is the space of distributions. The space $\Phi ^{\times}$ is called the antidual space of $\Phi$. Associated with the rigged Hilbert space~(\ref{RHStIntro}), there is always another rigged Hilbert space, \begin{equation} {\Phi} \subset {\cal H} \subset {\Phi}^{\prime} \, , \label{RHSpIntro} \end{equation} where ${\Phi}^{\prime}$ is called the dual space of ${\Phi}$ and contains the linear functionals over $\Phi$. The basic reason why we need the spaces ${\Phi}^{\prime}$ and ${\Phi}^{\times}$ is that the bras and kets associated with the elements in the continuous spectrum of an observable belong, respectively, to ${\Phi}^{\prime}$ and ${\Phi}^{\times}$ rather than to ${\cal H}$. The basic reason reason why we need the space $\Phi$ is that unbounded operators are not defined on the whole of ${\cal H}$ but only on dense subdomains of ${\cal H}$ that are not invariant under the action of the observables. Such non-invariance makes expectation values, uncertainties and commutation relations not well defined on the whole of $\cal H$. The space $\Phi$ is the largest subspace of the Hilbert space on which such expectation values, uncertainties and commutation relations are well defined. Besides accommodating resonances and Dirac's bra-ket formalism, the rigged Hilbert space seems to capture the physical principles of quantum mechanics better than the Hilbert space. For example, assuming that the Hilbert space provides the whole mathematical framework for quantum mechanics leads to the conclusion that Heisenberg's uncertainty relations are not physical, since they cannot be defined on the whole of the Hilbert space~\cite{PERES}. Using the rigged Hilbert space, one overcomes this difficulty after realizing that the commutation relations are well defined on $\Phi$. The completeness relation is a good place to appreciate the added value of the rigged Hilbert space. Consider, for example, the Hamiltonian $H$ of a system. In the Hilbert space, one writes the completeness relation as \begin{equation} 1=\int_{{\rm Sp}(H)} d{\sf E}_E \, , \end{equation} where ${\sf E}_E$ are the spectral projections of $H$ and ${\rm Sp}(H)$ is its spectrum. However, within the rigged Hilbert space one can write\footnote{In the Hilbert space, one can actually write something close to, although not the same as~(\ref{crHS}), by means of direct integral decompositions.} \begin{equation} 1= \sum_n |E_n\rangle \langle E_n| + \int_0^{\infty}dE \, |E\rangle \langle E| \, , \label{crHS} \end{equation} where $|E_n\rangle$ and $|E\rangle$ are the bound and scattering states of $H$, respectively. In addition to~(\ref{crHS}), the rigged Hilbert space gives you an additional completeness relation in which the resonance states participate: \begin{equation} 1= \sum_n |E_n\rangle \langle E_n| +\sum_n |z_n\rangle \langle z_n| + \int_{-\infty}^0dE \, |E\rangle \langle E| \, , \label{crRHS} \end{equation} where $|z_n\rangle$ are the Gamow (resonance) states of $H$ and the last integral, called the background, is performed in the complex plane right below the negative real axis of the second sheet.\footnote{As we will see in Lecture~5, the expansion~(\ref{crRHS}) must be either regulated or understood in a time-dependent way.} Thus, the completeness relation~\eqref{crRHS} substitutes the scattering states contribution by the resonance contribution plus a background, thereby putting the resonance spectrum on the same footing as the bound and scattering spectra. It is important to note that the integrals in (\ref{crHS}) and (\ref{crRHS}) are different, and that the resonance contribution does not appear in~(\ref{crHS}), because resonances are not asymptotic states. Also important is to note that the resonance states, and therefore expansion~(\ref{crRHS}), need a different rigged Hilbert space from that needed by the scattering states and expansion~(\ref{crHS}). There are dangers in using the rigged Hilbert space indiscriminately, though. For instance, A.~Bohm and collaborators have been using a rigged Hilbert space (of Hardy class) to construct a quantum theory of resonances, see review~\cite{IJTP03} and references therein. However, such theory is inconsistent with quantum mechanics and must be discarded~\cite{HARDY}. The structure of these lectures is as follows. After an introductory Lecture~1, I will explain in Lecture~2 how to construct the rigged Hilbert space of the one-dimensional rectangular barrier potential. This will show that the rigged Hilbert space is already needed to provide the mathematical support of the most basic quantum systems. In Lecture~3, I will construct the rigged Hilbert space of the Lippmann-Schwinger equation, which equation governs quantum scattering. In Lecture~4, I will construct the rigged Hilbert space of the analytic continuation of the Lippmann-Schwinger equation. The rigged Hilbert space of Lecture~4 will be needed in the final Lecture~5 to provide the mathematical support for the resonance states. The PDF file of each talk will be posted at \url{http://www.ucsd.edu/~rafa}. Since space prevents a full account, at each lecture I will refer the reader to the appropriate papers where further details can be found. The essentials of functional analysis needed to understand those papers can be found in~\cite[Chapter 2]{DIS}. \section{Construction of a simple rigged Hilbert space: Lecture~2} \label{sec:motivRHS} If resonances are not included, the way to construct the rigged Hilbert space of a quantum system is as follows: \begin{enumerate} \item We identify the observables of the system. Their expressions are usually given by linear, differential operators. \item We identify the Hilbert space, whose scalar product is used to calculate probability amplitudes. \item We identify the domains, spectra and eigenfunctions of the observables. If the observables have a discrete spectrum, we need not go beyond the Hilbert space. However, if at least one of the observables has continuous spectrum, we need to enlarge the Hilbert space to the rigged Hilbert space. (If position and momentum are among the observables, we will always need the rigged Hilbert space.) \item We construct the space $\Phi$ in which physical quantities such as expectation values, uncertainties and commutation relations are well defined. When resonances are not included, the space $\Phi$ is usually given by the maximal invariant subspace of the algebra of observables. \item We construct the dual $\Phi '$ and antidual $\Phi ^{\times}$ spaces. We construct the bras and kets of the observables and check that they respectively belong to $\Phi '$ and $\Phi ^{\times}$. \item The completeness relations and all the features of Dirac's bra-ket formalism now follow. \end{enumerate} Let's see how the above steps are carried out in the case of a spinless particle moving in one dimension and impinging on a rectangular barrier. The observables relevant to this system are the position $Q$, the momentum $P$, and the Hamiltonian $H$: \begin{equation} Qf(x)=xf(x) \, , \label{fdopx} \end{equation} \begin{equation} Pf(x)=-i \hbar \frac{d}{d x}f(x) \, , \label{fdopp} \end{equation} \begin{equation} Hf(x)= \left( -\frac{\hbar ^2}{2m}\frac{d ^2}{d x^2}+V(x) \right) f(x) \, , \label{fdoph} \end{equation} where \begin{equation} V(x)=\left\{ \begin{array}{ll} 0 &-\infty 0 \, , \end{equation} \begin{equation} \langle ^{\pm}k|r \rangle = \overline{\chi ^{\pm}(r;k)} = \chi ^{\mp}(r;k) \, , \quad k>0 \, . \label{leftLSe+-} \end{equation} Because of~(\ref{defiphi+-}), in terms of $k$ the Lippmann-Schwinger bras and kets read as \begin{eqnarray} \langle ^{\pm}k| = \sqrt{\frac{\hbar ^2}{2m}\, 2k\,} \, \langle ^{\pm}E| \, , \quad k>0 \, , \label{brask} \\ |k^{\pm}\rangle = \sqrt{\frac{\hbar ^2}{2m}\, 2k\,} \, |E^{\pm}\rangle \, , \quad k>0 \, . \label{ketsk} \end{eqnarray} The bras $\langle ^{\pm}k|$ and kets $|k^{\pm}\rangle$ are, respectively, left and right eigenvectors of $H$ with eigenvalue $\frac{\hbar ^2}{2m}k^2$: \begin{eqnarray} \langle ^{\pm}k| H =\frac{\hbar ^2}{2m}k^2 \langle ^{\pm}k| \, , \\ H |k^{\pm}\rangle =\frac{\hbar ^2}{2m}k^2 |k^{\pm}\rangle \, . \end{eqnarray} \subsection{The analytic continuation of the Lippmann-Schwinger eigenfunctions} The analytic continuation of $\chi ^{\pm}(r;k)$ is done as follows. First, one specifies the boundary values that the Lippmann-Schwinger eigenfunctions take on the positive $k$-axis. And second, one continues those boundary values into the whole $k$-plane. Since the boundary values of the Lippmann-Schwinger eigenfunctions on the positive $k$-axis are given by Eq.~(\ref{defiphi+-}), and since $\chi ^{\pm}(r;k)$ are expressed in terms of well-known analytic functions, the continuation of $\chi ^{\pm}(r;k)$ from the positive $k$-axis into the whole wave-number plane is well defined. A word on notation. Whenever they become complex, we will denote the energy $E$ and the wave number $k$ by respectively $z$ and $q$. Accordingly, the continuations of $\chi ^{\pm}(r;E)$ and $\chi ^{\pm}(r;k)$ will be denoted by $\chi ^{\pm}(r;z)$ and $\chi ^{\pm}(r;q)$. In bra-ket notation, the analytically continued eigenfunctions will be written as \begin{eqnarray} \langle r|q^{\pm}\rangle =\chi ^{\pm}(r;q) \, , \\ \langle ^{\pm}q|r \rangle =\chi ^{\mp}(r;q) \, . \end{eqnarray} Note that, in distinction to~(\ref{pmeigndubra}), the analytically continued ``left'' eigenfunction is not the complex conjugate of the ``right'' eigenfunction but \begin{equation} \langle ^{\pm}q|r \rangle =\overline{\langle r|\overline{q}^{\pm}\rangle} \, . \label{lrconus} \end{equation} In order to characterize the analytic properties of $\chi ^{\pm}(r;q)$, it is useful to define \begin{equation} Z_{\pm} \equiv \{ q \in {\mathbb C} \, | \ {\cal J}_{\pm}(q) = 0 \} \, . \end{equation} The elements of $Z_+$ are simply the resonance energies. Since $\chi (r;q)$ and ${\cal J}_{\pm}(q)$ are analytic in the whole $k$-plane, $\chi ^{\pm}(r;q)$ is analytic in the whole $k$-plane except at $Z_{\pm}$, where its poles are located. In order to define the analytically continued Lippmann-Schwinger bras and kets, we need to know how $\chi ^{\pm}(r;q)$ grow with $q$. To find out, let us recall first that the growth of $\chi (r;q)$ is bounded by~\cite{TAYLOR} \begin{equation} \left| \chi (r;q)\right| \leq C \, \frac{\left|q\right|r}{1+\left|q\right|r} \, e^{|{\rm Im}(q)|r} \, , \quad q\in {\mathbb C} \, . \label{boundrs} \end{equation} From Eqs.~(\ref{defiphi+-}) and (\ref{boundrs}), it follows that the eigenfunctions $\chi ^{\pm}(r;q)$ satisfy \begin{equation} \hskip-1cm \left| \chi ^{\pm}(r;q) \right| \leq \frac{C}{|{\cal J}_{\pm}(q)|} \, \frac{|q|r}{1+|q|r} \, e^{|{\rm Im}(q)|r } \, . \label{estimateofphi} \end{equation} When $q \in Z_{\pm}$, $\chi ^{\pm}(r;q)$ blows up to infinity. We can further refine the estimates~(\ref{estimateofphi}) by characterizing the growth of $1/|{\cal J}_{\pm}(q)|$ in different regions of the complex plane. In the upper half of the complex $k$-plane, the inverse of ${\cal J}_{+}(q)$ is bounded: \begin{equation} \frac{1}{\left| {\cal J}_{+}(q) \right|} \leq C \, , \quad {\rm Im}(q)\geq 0 \, . \label{boundinjslp+} \end{equation} In the lower half-plane, $\frac{1}{{\cal J}_{+}(q)}$ is infinite whenever $q \in Z_+$. As $|q|$ tends to $\infty$ in the lower half plane, we have \begin{equation} \frac{1}{{\cal J}_{+}(q)} \approx \frac{1}{1 - C q^{-2} e ^{2i q b}} \, , \quad (|q|\to \infty \, , \ {\rm Im}(q)<0) \, . \label{boundinjsup+} \end{equation} The above estimates are satisfied by ${\cal J}_{-}(q)$ when we exchange the upper for the lower half plane, and $Z_+$ for $Z_-$. \subsection{The analytic continuation of the Lippmann-Schwinger bras and kets} The analytic continuation of the Lippmann-Schwinger bras is defined for any complex wave number $q$ in the distributional way~(\ref{afGtff1}): \begin{equation} \langle ^{\pm}q|\varphi ^{\pm}\rangle \equiv \int_0^{\infty}d r\, \varphi ^{\pm} (r) \chi ^{\mp}(r;q) = \int_0^{\infty}d r\, \langle ^{\pm}q|r\rangle \langle r|\varphi ^{\pm}\rangle \, , \label{LSdefinitionbra+-q} \end{equation} where the functions $\varphi ^{\pm}(r)$ belong to a space of test functions ${\Phi}_{\rm exp}$ that will be constructed below. Similarly to the bras, the analytic continuation of the Lippmann-Schwinger kets is defined by way of~(\ref{afFtff1}): \begin{equation} \langle \varphi ^{\pm}|q^{\pm}\rangle \equiv \int_0^{\infty}d r\, \overline{\varphi ^{\pm} (r)} \chi ^{\pm}(r;q) = \int_0^{\infty}d r\, \langle \varphi ^{\pm}|r\rangle \langle r|q^{\pm}\rangle \, . \label{LSdefinitionket+-q} \end{equation} Note that definition~(\ref{LSdefinitionbra+-q}) is actually a slight generalization of~(\ref{afGtff1}), due to~(\ref{lrconus}). The bras~(\ref{LSdefinitionbra+-q}) and kets~(\ref{LSdefinitionket+-q}) are defined for all complex $q$ except at those $q$ at which the corresponding eigenfunction has a pole. At those poles, one can still define bras and kets if in definitions~(\ref{LSdefinitionbra+-q}) and (\ref{LSdefinitionket+-q}) one substitutes the eigenfunctions $\chi ^{\pm}(r;q)$ by their residues at the pole. From the analytic continuation of the bras and kets into any complex wave number, one can now obtain the analytic continuation of the bras and kets into any complex energy of the Riemann surface (compare with Eqs.~(\ref{brask}) and (\ref{ketsk})): \begin{equation} |z^{\pm}\rangle =\sqrt{\frac{2m}{\hbar ^2}\frac{1}{2q}}\, |q^{\pm}\rangle \, , \quad \langle ^{\pm}z|=\sqrt{\frac{2m}{\hbar ^2}\frac{1}{2q}}\, \langle ^{\pm}q| \, . \end{equation} \subsection{Construction of the rigged Hilbert space} Likewise the bras and kets associated with real energies, the analytic continuation of the Lippmann-Schwinger bras and kets must be described within the rigged Hilbert space rather than just within the Hilbert space. We will denote the rigged Hilbert space for the analytically continued bras by \begin{equation} {\Phi}_{\rm exp} \subset L^2([0,\infty ),d r) \subset {\Phi}_{\rm exp}^{\prime} \, , \label{rhsexpp} \end{equation} and the one for the analytically continued kets by \begin{equation} {\Phi}_{\rm exp} \subset L^2([0,\infty ),dr) \subset {\Phi}_{\rm exp}^{\times} \, . \label{rhsexpt} \end{equation} The functions $\varphi ^{\pm}\in {\Phi}_{\rm exp}$ must satisfy the following conditions: \begin{eqnarray} \hskip-2cm && \bullet \ \mbox{They belong to the maximal invariant subspace $\cal D$ of} \ H, \ \mbox{see Eq.~(\ref{misoH}).} \label{condition1exp} \\ [2ex] \hskip-2cm &&\bullet \ \mbox{They are such that definitions~(\ref{LSdefinitionbra+-q}) and (\ref{LSdefinitionket+-q}) make sense.} \label{condition2exp} \end{eqnarray} The reason why $\varphi ^{\pm}$ must satisfy condition~(\ref{condition1exp}) is that such condition guarantees that all the powers of the Hamiltonian are well defined. Condition~(\ref{condition1exp}), however, is not sufficient to obtain well-defined bras and kets associated with complex wave numbers. In order for $\langle ^{\pm}q|$ and $|q^{\pm}\rangle$ to be well defined, the wave functions $\varphi ^{\pm}(r)$ must be well behaved so the integrals in Eqs.~(\ref{LSdefinitionbra+-q}) and (\ref{LSdefinitionket+-q}) converge. Since by Eq.~(\ref{estimateofphi}) $\chi ^{\pm}(r;q)$ grow exponentially with $r$, the wave functions $\varphi ^{\pm}(r)$ have to, essentially, tame real exponentials. If we define \begin{equation} \| \varphi ^{\pm}\|_{n,n'} \equiv \sqrt{\int_{0}^{\infty}d r \, \left| \frac{nr}{1+nr}\, e ^{nr^2/2} (1+H)^{n'} \varphi ^{\pm}(r) \right|^2 \, } \, , \quad n,n'=0,1,2, \ldots \, , \label{normsLSexp} \end{equation} then the space ${\Phi}_{\rm exp}$ is given by \begin{equation} {\Phi}_{\rm exp} = \left\{ \varphi ^{\pm}\in {\cal D} \, | \ \| \varphi ^{\pm} \|_{n,n'}<\infty \, , \ n,n'=0,1,2,\ldots \right\} . \label{phiexp} \end{equation} This is just the space of square integrable functions which belong to the maximal invariant subspace of $H$ and for which the quantities~(\ref{normsLSexp}) are finite. In particular, because $\varphi ^{\pm}(r)$ satisfy the estimates~(\ref{normsLSexp}), their tails fall off faster than Gaussians. From Eq.~(\ref{estimateofphi}), it is clear that the integrals in Eqs.~(\ref{LSdefinitionbra+-q}) and (\ref{LSdefinitionket+-q}) converge already for functions that fall off at infinity faster than any exponential. We have imposed Gaussian falloff because it will allow us to perform resonance expansions in Lecture~5. It is illuminating to compare the space $\Phi$ of Lecture~3 with the space ${\Phi}_{\rm exp}$ of Eq.~(\ref{phiexp}). Because for real wave numbers the Lippmann-Schwinger eigenfunctions behave like purely imaginary exponentials, in Lecture~3 we only needed to impose on the test functions a polynomial falloff, thereby obtaining a space of test functions very similar to the Schwartz space. By contrast, for complex wave numbers the Lippmann-Schwinger eigenfunctions blow up exponentially, and therefore we need to impose on the test functions an exponential falloff that damps such an exponential blowup. One can now easily show that the kets $|q^{\pm}\rangle$ belong to ${\Phi}_{\rm exp}^{\times}$ and satisfy \begin{equation} H|q^{\pm}\rangle= \frac{\hbar ^2}{2m}q^2 \, |q^{\pm}\rangle \, , \label{keigeeqa} \end{equation} \begin{equation} e^{-i Ht/\hbar}|q^{\pm}\rangle = e^{-i q^2 \hbar t/(2m)} |q^{\pm}\rangle \, . \label{timevoketspm1} \end{equation} Similarly, $\langle ^{\pm}q| \in {\Phi}_{\rm exp}^{\prime}$ and \begin{equation} \langle ^{\pm}q|H=\frac{\hbar ^2}{2m} q^2 \langle ^{\pm}q|\, , \label{kpssleftkeofHa} \end{equation} \begin{equation} \langle ^{\pm}q|e^{-i Ht/\hbar} = e^{i q^2 \hbar t/(2m)} \langle ^{\pm}q| \, . \label{timevobraspm2} \end{equation} Equations~(\ref{keigeeqa}) and (\ref{kpssleftkeofHa}) can be rewritten in terms of the complex energy $z$ as \begin{eqnarray} H|z^{\pm}\rangle=z|z^{\pm}\rangle \, , \label{eigenzi} \\ [1ex] \langle ^{\pm}z|H = z \langle ^{\pm}z| \, . \label{eigenzibra} \end{eqnarray} Note that~(\ref{eigenzibra}) is not given by $\langle ^{\pm}z|H = \overline{z}\langle ^{\pm}z|$, as one may naively expect from formally obtaining~(\ref{eigenzibra}) by Hermitian conjugation of~(\ref{eigenzi}). For a full account of this lecture, the reader can refer to~\cite{LS2}. \section{Resonance states, and their rigged Hilbert space: Lecture~5} \label{sec:RHSresonan} The Gamow states are the state vectors of resonances. They are eigenvectors of the Hamiltonian with a complex eigenvalue. The real (imaginary) part of the complex eigenvalue is associated with the energy (width) of the resonance. Because self-adjoint operators on a Hilbert space can only have real eigenvalues, the Gamow states fit not within the Hilbert space but within the rigged Hilbert space. In this lecture, we will see that the Gamow states belong to the rigged Hilbert space of Lecture~4. Like in Lectures~3 and~4, we will use the spherical shell potential~(\ref{sbpotential}) and study the zero angular momentum case only. Unlike in Lecture~4, we will write most results in terms of the energy, because they tend to be simpler than in terms of the wave number. The energy and the wave number of a resonance ${\rm R}$ will be denoted by $z_{\rm R}$ and $k_{\rm R}$. The Gamow eigenfunctions satisfy the Schr\"odinger equation \begin{equation} \left( -\frac{\hbar ^2}{2m}\frac{d ^2}{d r^2}+V(r)\right) u(r;z_{\rm R}) = z_{\rm R} \, u(r;z_{\rm R}) \, , \label{Grse0} \end{equation} subject to ``purely outgoing boundary conditions,'' \begin{eqnarray} && u(0;z_{\rm R}) = 0 \, , \label{gvlov1} \\ && u(r;z_{\rm R}) \ \mbox{is continuous at} \ r=a,b \, , \label{gvlov2} \\ &&\frac{d}{d r}u(r;z_{\rm R}) \ \mbox{is continuous at} \ r=a,b \, , \label{gvlov5} \\ && u(r;z_{\rm R}) \sim e^{i k_{\rm R}r} \ \mbox{as} \ r\to \infty \, , \label{gvlov6} \end{eqnarray} where~(\ref{gvlov6}) is the ``purely outgoing boundary condition'' (POBC). Comparison of (\ref{gvlov1})-(\ref{gvlov6}) with (\ref{bcoLS1})-(\ref{bcoLS6}) shows that it is the POBC what selects the resonance energies. For the potential~(\ref{sbpotential}), the only possible eigenvalues of~(\ref{Grse0}) subject to~(\ref{gvlov1})-(\ref{gvlov6}) are the zeros of the Jost function, \begin{equation} {\cal J}_+(z_{\rm R})=0 \, . \label{ressoncon} \end{equation} The solutions of this equation come as a denumerable number of complex conjugate pairs $z_n, z_n^*$. The number $z_n=E_n -i \Gamma _n /2$ is the $n$th resonance energy, and $z_n^*=E_n +i \Gamma _n /2$ is the $n$th anti-resonance energy. The corresponding wave numbers are \begin{equation} k_n=\sqrt{\frac{2m}{\hbar ^2}z_n\,} \, , \quad -k_n^*=\sqrt{\frac{2m}{\hbar ^2}z_n^*\,} \, , \quad n=1,2, \ldots \, . \end{equation} For the potential~(\ref{sbpotential}), the resonance energies are simple poles of the $S$ matrix (see~\cite{MONDRADOUBLE} for an example of a potential that produces double poles). In order to write expressions for resonances and anti-resonances together, we will label the resonances by a positive integer $n=1,2,\ldots$ and the anti-resonances by a negative integer $n=-1,-2,\ldots$. The $n$th Gamow eigensolution, $n=\pm 1, \pm 2, \ldots$, reads \begin{equation} u(r;z_n)=u(r;k_n)= N_n\left\{ \begin{array}{ll} \frac{1}{{\mathcal J}_3(k_n)}\sin(k_{n}r) &00 \, , \ n=1,2, \ldots \, , \label{adjoiotev2ares} \end{equation} whereas for anti-resonances, it should be that \begin{equation} \langle z_n| e^{-i Ht/\hbar}= e^{i z_n t/\hbar} \langle z_n| \, , \quad {\rm only\ for}\ t>0 \, , \ n=-1,-2, \ldots \, , \label{adjoiotevbra1aresta} \end{equation} \begin{equation} e^{-i Ht/\hbar}|z_n\rangle = e^{-i z_n t/\hbar} |z_n\rangle \, , \quad {\rm only\ for}\ t<0 \, , \ n=-1,-2, \ldots \, . \label{adjoiotev2aresta} \end{equation} If we define the complex delta function at $z_n$ by \begin{equation} \int_0^{\infty}dE \, f(E) \delta (E-z_n) = f(z_n) \, , \end{equation} one can use the results of Lecture~4 to show that the Gamow eigenfunction $u(r;z_n)$, the complex delta function (multiplied by a normalization factor) and the Breit-Wigner amplitude (multiplied by a normalization factor) are linked with each other: \begin{equation} \begin{array}{ccccc} u(r;z_n)& \leftrightarrow & i \sqrt{2\pi \, } {\cal N}_n \delta (E-z_n) \, , \ E\in [0,\infty ) & \leftrightarrow & -\frac{{\cal N}_n}{\sqrt{2\pi}}\, \frac{1}{E-z_n}\, , \ E\in (-\infty ,\infty) \\ [2ex] \mbox{posit. repr.} &\ & \mbox{energy repr.} & \ & (-\infty,\infty)\mbox{-``energy'' repr.} \end{array} \label{greatres} \end{equation} where \begin{equation} {\cal N}_n^2 = i \, {\rm res}[S(z)]_{z=z_n} \, . \end{equation} Physically, these links mean that the Gamow states yield a decay amplitude ${\cal A}(z_{\rm R}\to E)$ given by the complex delta function, and that such decay amplitude can be approximated by the Breit-Wigner amplitude when we can ignore the lower bound of the energy, i.e., when the resonance is so far from the threshold that we can safely assume that the energy runs over the full real line: \begin{equation} {\cal A}(z_{n}\to E)= \langle ^-E|z_{n}\rangle = i \sqrt{2\pi} {\cal N}_{n} \delta (E-z_{n}) \simeq -\frac{{\cal N}_{n}}{\sqrt{2\pi}} \, \frac{1}{E-z_{n}} \, . \label{mainresult} \end{equation} Thus, the almost-Lorentzian peaks in cross sections are caused by intermediate, unstable particles. However, because there is actually a lower bound for the energy, the decay amplitude is never exactly given by the Breit-Wigner amplitude. This means, in particular, that the standard Gamow states are different from the so-called ``Gamow vectors'' of~\cite{IJTP03}. \subsection{Resonance expansions} The scattering bras and kets are basis vectors that furnish the completeness relation~(\ref{crHS}). The Gamow states are also basis vectors. The completeness relation~(\ref{crRHS}) generated by the Gamow states is called the resonance expansion. Resonance expansions are almost always obtained in the same way. One starts from the expansion in terms of bound and scattering states and then, by deforming the continuum integral into the complex plane, and by Cauchy's theorem, one extracts the contributions from the resonances that are hidden in the continuum and write them in the same way as the contributions from the bound states. For the sake of simplicity, we will focus on the resonance expansion of the transition amplitude from an ``in'' state $\varphi ^+$ into an ``out'' state $\varphi ^-$: \begin{equation} \left( \varphi ^-,\varphi ^+\right)= \int_0^{\infty} d E \, \langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle \, . \label{superequation} \end{equation} We now extract the resonance contributions out of~(\ref{superequation}) by deforming the contour of integration into the lower half plane of the second sheet of the Riemann surface, where the resonance poles are located, and by applying Cauchy's theorem. Assuming that the integrand $\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ tends to zero in the infinite arc of the lower half plane of the second sheet, the resulting resonance expansion is \begin{equation} \left( \varphi ^-,\varphi ^+\right) = \sum_{n=1}^{\infty} \langle \varphi ^-|z_n\rangle\langle z_n|\varphi ^+\rangle + \int_0^{-\infty} d E \, \langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle \, . \label{zigzag} \end{equation} The integral in Eq.~(\ref{zigzag}) is supposed to be done infinitesimally below the negative real semiaxis of the second sheet. By omitting the wave functions in~(\ref{zigzag}), we obtain the completeness relation~(\ref{crRHS}). In Eq.~(\ref{zigzag}), the infinite sum contains the contribution from the resonances, while the integral is the non-resonant background. Note that in~(\ref{zigzag}) bound states do not appear, since the potential~(\ref{sbpotential}) doesn't bind any. In obtaining Eq.~(\ref{zigzag}), we have assumed that the integrand $\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ tends to zero in the infinite arc of the lower half plane of the second sheet. Since~(\ref{zigzag}) makes physical sense, we are tempted to conclude that such must be the case. However, as showed in~\cite{LS2}, $\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ does not tend to zero in the infinite arc of the lower half plane of the second sheet. On the contrary, it diverges exponentially there. Therefore, Eq.~(\ref{zigzag}) doesn't make sense as it stands. In order to make sense of it, one has to control the exponential blowup of $\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$. The way to do so is by introducing an exponentially damping regulator $e^{-i \alpha z}$, $\alpha >0$. Thus, Eq.~(\ref{zigzag}) should read \begin{equation} \left( \varphi ^-,\varphi ^+\right) = \lim_{\alpha \to 0} \sum_{n=1}^{\infty} e^{-i \alpha z_n} \langle \varphi ^-|z_n \rangle\langle z_n|\varphi ^+\rangle + \int_0^{-\infty} d E \, e^{-i \alpha E} \langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle . \label{alphazigzag} \end{equation} Physically, the regulator $e^{-i \alpha z}$ is simply the analytic continuation of the time evolution operator in the energy representation, $e^{-i z \alpha} \equiv e^{-i zt/\hbar}$. Thus, the above regularized equation must be understood in a time-asymmetric, time-dependent fashion as \begin{equation} (\varphi ^-, e^{-i H\frac{t}{\hbar}} \varphi ^+ ) = \sum_{n=1}^{\infty} e^{-i z_n\frac{t}{\hbar}} \langle \varphi ^-|z_n\rangle \langle z_n|\varphi ^+\rangle + \int_0^{-\infty} dE \, e^{-i E\frac{t}{\hbar}} \langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle \label{alphazigzagtime} \end{equation} for $t>0$ only. Equation~(\ref{zigzag}) should then be seen as the (singular) limit of Eq.~(\ref{alphazigzagtime}) when $t \to 0^+$. That for resonances $t \equiv \alpha \hbar$ must be positive is in accord with the time asymmetry of~(\ref{adjoiotev2ares}). Expansions~(\ref{alphazigzag}) and (\ref{alphazigzagtime}) are the reason why we chose a Gaussian falloff for the elements of $\Phi _{\rm exp}$: When the wave functions have a Gaussian falloff in the position representation, we can regularize their blowup in the energy representation and interpret the regulator as a time-asymmetric evolution. Resonance expansions allow us to understand the deviations from exponential decay. When a particular resonance, say resonance 1, is dominant, then the Gamow state of resonance~1 will carry the exponential decay, whereas the background, which includes in this case also the contribution from other possible resonances, carries the deviations from exponential decay. The full account of this lecture will appear in a forthcoming paper. \section{Conclusions} \label{sec:conclusions} We have seen why the rigged Hilbert space, rather than the Hilbert space alone, is needed to formulate quantum mechanics when the observables have continuous and/or resonance spectra. The rigged Hilbert space captures the physics of continuous and resonance spectra better than the Hilbert space, because in the rigged Hilbert space physical quantities such as commutation relations, uncertainty principles and resonances have always a precise meaning. In addition to provide the mathematical support for Dirac's bra-ket formalism, for the Lippmann-Schwinger equation and for the Gamow states, the rigged Hilbert space can be used to obtain the resonance (decay) amplitude in terms of the complex delta function. Such decay amplitude can be approximated by the Breit-Wigner amplitude when the lower bound of the energy can be ignored. To finish, I would like to mention that there is still a long list of pending questions worth pursuing, such as the invariance properties of $\Phi _{\rm exp}$ under time evolution or a detailed proof of the asymmetry in the time evolution of the Gamow states. \begin{theacknowledgments} It is a pleasure to thank the organizers for their invitation to participate in this summer school and Oscar Rosas-Ortiz for his hospitality. This work was supported by MEC fellowship No.~SD2004-0003. \end{theacknowledgments} \begin{thebibliography}{99} \bibitem{PERES} A.~Peres, \emph{Quantum Theory: Concepts and Methods}, Dordrecht, Kluwer Academic (1993). \bibitem{IJTP03} A.~Bohm, Int.~J.~Theo.~Phys.~{\bf 42}, 2317 (2003). \bibitem{HARDY} R.~de la Madrid, J.~Phys.~A: Math.~Gen.~{\bf 39}, 9255 (2006); {\sf quant-ph/0606186}. \bibitem{DIS} R.~de la Madrid, \emph{Quantum Mechanics in Rigged Hilbert Space Language}, PhD Dissertation, Universidad de Valladolid, Spain (2001). Available at \url{http://www.physics.ucsd.edu/~rafa}. \bibitem{04JPA} R.~de la Madrid, J.~Phys.~A: Math.~Gen.~{\bf 37}, 8129 (2004); {\sf quant-ph/0407195}. \bibitem{05EJP} R.~de la Madrid, Eur.~J.~Phys.~{\bf 26}, 287 (2005); {\sf quant-ph/0502053}. \bibitem{LS1} R.~de la Madrid, J.~Phys.~A:~Math.~Gen.~{\bf 39}, 3949 (2006); {\sf quant-ph/0603176}. \bibitem{TAYLOR} J.~R.~Taylor, \emph{Scattering theory}, John Wiley \& Sons, Inc., New York (1972). \bibitem{LS2} R.~de la Madrid, J.~Phys.~A:~Math.~Gen.~{\bf 39}, 3981 (2006); {\sf quant-ph/0603177}. \bibitem{MONDRADOUBLE} E.~Hern\'andez, A.~J\'auregui, A.~Mondrag\'on, J.~Phys.~A: Math.~Gen.~{\bf 33}, 4507 (2000). \end{thebibliography} \end{document} ---------------0607241637508--