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y(where)27 b(for)g Fm(\020)i Fl(2)24 b Fj(R)p Fn(,)1465 3908 y Fm(T)1514 3920 y Fi(E)1570 3908 y Fn(\()p Fm(\020)6 b Fn(\))24 b(=)1787 3791 y Fh(\022)1890 3857 y Fm(E)g Fl(\000)18 b Fm(\020)89 b Fl(\000)p Fn(1)1974 3957 y(1)199 b(0)2331 3791 y Fh(\023)2406 3908 y Fm(:)515 4103 y Fn(Then,)28 b(an)n(y)e(solution)h Fm(\036)i Fn(of)e(\(2\))h(ob)r (eys)1584 4279 y(\010\()p Fm(n)p Fn(\))c(=)e Fm(M)1950 4291 y Fi(E)2006 4279 y Fn(\()p Fm(n)p Fn(\)\010\(0\))p Fm(;)515 4455 y Fn(where)38 b(\010\()p Fm(i)p Fn(\))h(denotes)f(\()p Fm(\036)p Fn(\()p Fm(i)27 b Fn(+)e(1\))p Fm(;)14 b(\036)p Fn(\()p Fm(i)p Fn(\)\))1813 4425 y Fi(T)1866 4455 y Fn(.)70 b(The)39 b(main)g(result)f(of)h(this)g(section)f(is)h(the)515 4555 y(follo)n(wing:)515 4731 y Fg(Theorem)30 b(1)41 b Ff(Fix)30 b(some)g Fm(E)f Fl(2)23 b Fj(R)p Ff(.)45 b(Supp)l(ose)30 b(that)593 4907 y Fn(\(i\))42 b Ff(Ther)l(e)d(exists)e (a)h(se)l(quenc)l(e)g(of)g(inte)l(gers)g Fm(n)2102 4919 y Fi(k)2180 4907 y Fl(!)g(1)f Ff(such)h(that)g(for)g(every)h Fm(k)h Ff(and)722 5006 y(every)31 b Fn(1)22 b Fl(\024)h Fm(i)g Fl(\024)f(b)p Fm(n)1320 5018 y Fi(k)1361 5006 y Fm(=)p Fn(2)p Fl(c)p Ff(,)29 b(we)h(have)h Fm(V)19 b Fn(\()p Fm(i)p Fn(\))k(=)g Fm(V)c Fn(\()p Fm(n)2270 5018 y Fi(k)2329 5006 y Fl(\000)f Fm(i)g Fn(+)g(1\))p Ff(.)1926 5255 y Fn(2)p eop %%Page: 3 3 3 2 bop 570 523 a Fn(\(ii\))42 b Ff(Ther)l(e)29 b(exists)f(a)h(c)l (onstant)e Fm(C)1644 535 y Fk(1)1710 523 y Ff(such)i(that)f Fl(j)p Fn(tr)p Fm(M)2235 535 y Fi(E)2290 523 y Fn(\()p Fm(n)2372 535 y Fi(k)2413 523 y Fn(\))p Fl(j)c(\024)e Fm(C)2638 535 y Fk(1)2704 523 y Ff(for)29 b(in\014nitely)g(many)722 623 y Fm(n)772 635 y Fi(k)813 623 y Ff(.)547 785 y Fn(\(iii\))42 b Ff(Ther)l(e)29 b(exists)f(a)h(c)l(onstant)e Fm(C)1644 797 y Fk(2)1710 785 y Ff(such)i(that)f Fl(j)p Fn(tr)p Fm(M)2235 797 y Fi(E)2290 785 y Fn(\()p Fm(n)2372 797 y Fi(k)2413 785 y Fn(\))p Fl(j)c(\025)e Fm(C)2638 797 y Fk(2)2704 785 y Ff(for)29 b(in\014nitely)g(many)722 885 y Fm(n)772 897 y Fi(k)813 885 y Ff(.)515 1058 y(Then)k(no)f (solution)g Fm(\036)h Ff(of)g(\(2\))f(tends)g(to)g Fn(0)g Ff(at)g Fn(+)p Fl(1)p Ff(.)45 b(In)32 b(p)l(articular,)i(no)e(solution) h(of)g(\(2\))515 1158 y(is)d(squar)l(e-summable)g(and)g(henc)l(e)g Fm(E)35 b Ff(is)30 b(not)f(an)h(eigenvalue)h(of)g Fm(H)7 b Ff(.)515 1332 y(R)l(emark.)62 b Fn(In)35 b(other)h(w)n(ords,)g(if)g (the)g(p)r(oten)n(tial)g(restricted)f(to)h(the)g(righ)n(t)f(half-line)h (has)515 1431 y(in\014nitely)41 b(man)n(y)g(palindromic)f(pre\014xes)h (and)g(the)g(traces)f(of)h(the)h(transfer)e(matrices)515 1531 y(asso)r(ciated)28 b(with)i(these)g(palindromes)e(neither)i(tend)g (to)g(0)f(nor)g(to)g Fl(1)h Fn(for)f(some)g(energy)515 1630 y Fm(E)5 b Fn(,)26 b(then)g(the)h(solutions)e(corresp)r(onding)e (to)j(this)g(energy)f(do)g(not)h(tend)g(to)g(0)f(at)h(+)p Fl(1)f Fn(and)515 1730 y(hence)i Fm(E)33 b Fn(is)28 b(not)f(an)g(eigen) n(v)-5 b(alue)27 b(of)h Fm(H)7 b Fn(.)515 1876 y Ff(Pr)l(o)l(of.)39 b Fn(W)-7 b(e)29 b(will)f(\014rst)f(sho)n(w)g(that)i(for)e(ev)n(ery)g Fm(k)s Fn(,)h(the)g(transfer)f(matrix)g Fm(M)2876 1888 y Fi(E)2932 1876 y Fn(\()p Fm(n)3014 1888 y Fi(k)3055 1876 y Fn(\))h(has)g(the)515 1975 y(form)1489 2204 y Fm(M)1570 2216 y Fi(E)1625 2204 y Fn(\()p Fm(n)1707 2216 y Fi(k)1748 2204 y Fn(\))23 b(=)1891 2087 y Fh(\022)1994 2153 y Fm(a)2038 2165 y Fi(k)2162 2153 y Fl(\000)p Fm(b)2263 2165 y Fi(k)1998 2253 y Fm(b)2034 2265 y Fi(k)2190 2253 y Fm(d)2233 2265 y Fi(k)2344 2087 y Fh(\023)3273 2204 y Fn(\(3\))515 2393 y(for)h(suitable)i(n)n(um)n(b)r(ers)e Fm(a)1323 2405 y Fi(k)1364 2393 y Fm(;)14 b(b)1437 2405 y Fi(k)1477 2393 y Fm(;)g(d)1557 2405 y Fi(k)1598 2393 y Fn(.)37 b(W)-7 b(e)25 b(consider)f(\014rst)h(the)h(case)e(where)h Fm(n)2890 2405 y Fi(k)2956 2393 y Fn(is)g(ev)n(en.)36 b(Let)1676 2612 y Fm(T)e Fn(=)1847 2495 y Fh(\022)1949 2561 y Fn(0)83 b(1)1949 2661 y(1)g(0)2157 2495 y Fh(\023)515 2835 y Fn(and)27 b(denote)h(the)g(matrix)f(en)n(tries)g(of)g Fm(M)1798 2847 y Fi(E)1854 2835 y Fn(\()p Fm(n)1936 2847 y Fi(k)1977 2835 y Fm(=)p Fn(2\))g(b)n(y)1451 3059 y Fm(M)1532 3071 y Fi(E)1587 3059 y Fn(\()p Fm(n)1669 3071 y Fi(k)1710 3059 y Fm(=)p Fn(2\))c(=)1936 2942 y Fh(\022)2039 3008 y Fm(\013)2092 3020 y Fi(k)2216 3008 y Fm(\014)2263 3020 y Fi(k)2044 3108 y Fm(\015)2087 3120 y Fi(k)2221 3108 y Fm(\016)2258 3120 y Fi(k)2345 2942 y Fh(\023)2420 3059 y Fm(:)639 3282 y Fn(Then,)28 b(using)f(assumption)g(\(i\),)i(w)n (e)e(get)995 3555 y Fm(M)1076 3567 y Fi(E)1131 3555 y Fn(\()p Fm(n)1213 3567 y Fi(k)1254 3555 y Fn(\))95 b(=)f Fm(M)1621 3567 y Fi(E)1677 3555 y Fn(\()p Fm(n)1759 3567 y Fi(k)1800 3555 y Fm(=)p Fn(2\))17 b Fl(\001)i Fm(T)29 b Fl(\001)19 b Fn(\()p Fm(M)2208 3567 y Fi(E)2264 3555 y Fn(\()p Fm(n)2346 3567 y Fi(k)2387 3555 y Fm(=)p Fn(2\)\))2535 3521 y Fe(\000)p Fk(1)2642 3555 y Fl(\001)g Fm(T)1381 3730 y Fn(=)1540 3613 y Fh(\022)1643 3679 y Fm(\013)1696 3691 y Fi(k)1820 3679 y Fm(\014)1867 3691 y Fi(k)1648 3779 y Fm(\015)1691 3791 y Fi(k)1825 3779 y Fm(\016)1862 3791 y Fi(k)1949 3613 y Fh(\023)2029 3730 y Fl(\001)f Fm(T)30 b Fl(\001)2190 3613 y Fh(\022)2328 3679 y Fm(\016)2365 3691 y Fi(k)2524 3679 y Fl(\000)p Fm(\014)2636 3691 y Fi(k)2293 3779 y Fl(\000)p Fm(\015)2401 3791 y Fi(k)2553 3779 y Fm(\013)2606 3791 y Fi(k)2718 3613 y Fh(\023)2798 3730 y Fl(\001)18 b Fm(T)1381 3962 y Fn(=)1540 3845 y Fh(\022)1643 3911 y Fm(\013)1696 3923 y Fi(k)1820 3911 y Fm(\014)1867 3923 y Fi(k)1648 4011 y Fm(\015)1691 4023 y Fi(k)1825 4011 y Fm(\016)1862 4023 y Fi(k)1949 3845 y Fh(\023)2029 3962 y Fl(\001)2070 3845 y Fh(\022)2202 3911 y Fm(\013)2255 3923 y Fi(k)2408 3911 y Fl(\000)p Fm(\015)2516 3923 y Fi(k)2173 4011 y Fl(\000)p Fm(\014)2285 4023 y Fi(k)2443 4011 y Fm(\016)2480 4023 y Fi(k)2598 3845 y Fh(\023)1381 4195 y Fn(=)1540 4077 y Fh(\022)1723 4144 y Fm(\013)1776 4114 y Fk(2)1776 4167 y Fi(k)1836 4144 y Fl(\000)g Fm(\014)1970 4114 y Fk(2)1966 4167 y Fi(k)2171 4144 y Fl(\000)p Fn(\()p Fm(\013)2321 4156 y Fi(k)2361 4144 y Fm(\015)2404 4156 y Fi(k)2464 4144 y Fl(\000)g Fm(\014)2594 4156 y Fi(k)2635 4144 y Fm(\016)2672 4156 y Fi(k)2712 4144 y Fn(\))1643 4244 y Fm(\013)1696 4256 y Fi(k)1737 4244 y Fm(\015)1780 4256 y Fi(k)1839 4244 y Fl(\000)g Fm(\014)1969 4256 y Fi(k)2010 4244 y Fm(\016)2047 4256 y Fi(k)2293 4244 y Fl(\000)p Fm(\015)2406 4213 y Fk(2)2401 4267 y Fi(k)2461 4244 y Fn(+)g Fm(\016)2584 4213 y Fk(2)2581 4267 y Fi(k)2786 4077 y Fh(\023)1369 4427 y Fn(=:)1540 4310 y Fh(\022)1643 4376 y Fm(a)1687 4388 y Fi(k)1810 4376 y Fl(\000)p Fm(b)1911 4388 y Fi(k)1647 4476 y Fm(b)1683 4488 y Fi(k)1839 4476 y Fm(d)1882 4488 y Fi(k)1993 4310 y Fh(\023)2068 4427 y Fm(:)515 4650 y Fn(This)27 b(pro)n(v)n(es)f(\(3\))i(for)f Fm(n)1272 4662 y Fi(k)1340 4650 y Fn(ev)n(en.)639 4750 y(Let)h(us)g(no)n(w)f (consider)f(the)i(case)f(where)g Fm(n)2000 4762 y Fi(k)2068 4750 y Fn(is)h(o)r(dd.)37 b(W)-7 b(riting)1414 4969 y Fm(M)1495 4981 y Fi(E)1550 4969 y Fn(\()p Fl(b)p Fm(n)1669 4981 y Fi(k)1710 4969 y Fm(=)p Fn(2)p Fl(c)p Fn(\))22 b(=)1973 4852 y Fh(\022)2076 4918 y Fm(\013)2129 4930 y Fi(k)2253 4918 y Fm(\014)2300 4930 y Fi(k)2081 5018 y Fm(\015)2124 5030 y Fi(k)2258 5018 y Fm(\016)2295 5030 y Fi(k)2382 4852 y Fh(\023)2457 4969 y Fm(;)1926 5255 y Fn(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fn(w)n(e)27 b(can)g(pro)r(ceed)g(with)h Fm(\020)i Fn(=)22 b Fm(V)d Fn(\()p Fl(b)p Fm(n)1624 535 y Fi(k)1665 523 y Fm(=)p Fn(2)p Fl(c)e Fn(+)i(1\))27 b(as)g(follo)n(ws:)691 805 y Fm(M)772 817 y Fi(E)828 805 y Fn(\()p Fm(n)910 817 y Fi(k)951 805 y Fn(\))95 b(=)f Fm(M)1318 817 y Fi(E)1374 805 y Fn(\()p Fl(b)p Fm(n)1493 817 y Fi(k)1534 805 y Fm(=)p Fn(2)p Fl(c)p Fn(\))17 b Fl(\001)i Fm(T)1795 817 y Fi(E)1850 805 y Fn(\()p Fm(\020)6 b Fn(\))20 b Fl(\001)e Fm(T)30 b Fl(\001)18 b Fn(\()p Fm(M)2250 817 y Fi(E)2306 805 y Fn(\()p Fl(b)p Fm(n)2425 817 y Fi(k)2466 805 y Fm(=)p Fn(2)p Fl(c)p Fn(\)\))2651 771 y Fe(\000)p Fk(1)2758 805 y Fl(\001)h Fm(T)1078 980 y Fn(=)1237 863 y Fh(\022)1340 929 y Fm(\013)1393 941 y Fi(k)1517 929 y Fm(\014)1564 941 y Fi(k)1345 1029 y Fm(\015)1388 1041 y Fi(k)1522 1029 y Fm(\016)1559 1041 y Fi(k)1646 863 y Fh(\023)1726 980 y Fl(\001)1767 863 y Fh(\022)1870 929 y Fm(E)k Fl(\000)18 b Fm(\020)90 b Fl(\000)p Fn(1)1954 1029 y(1)199 b(0)2310 863 y Fh(\023)2390 980 y Fl(\001)2431 863 y Fh(\022)2563 929 y Fm(\013)2616 941 y Fi(k)2769 929 y Fl(\000)p Fm(\015)2877 941 y Fi(k)2534 1029 y Fl(\000)p Fm(\014)2646 1041 y Fi(k)2805 1029 y Fm(\016)2842 1041 y Fi(k)2959 863 y Fh(\023)1078 1212 y Fn(=)1237 1095 y Fh(\022)1377 1161 y Fn(\()p Fm(E)23 b Fl(\000)18 b Fm(\020)6 b Fn(\))p Fm(\013)1703 1131 y Fk(2)1703 1185 y Fi(k)1763 1161 y Fn(+)19 b(2)p Fm(\013)1942 1173 y Fi(k)1982 1161 y Fm(\014)2029 1173 y Fi(k)2190 1161 y Fl(\000)p Fn(\(\()p Fm(E)k Fl(\000)18 b Fm(\020)6 b Fn(\))p Fm(\013)2613 1173 y Fi(k)2655 1161 y Fm(\015)2698 1173 y Fi(k)2757 1161 y Fn(+)18 b(2)p Fm(\014)2929 1173 y Fi(k)2970 1161 y Fm(\015)3013 1173 y Fi(k)3053 1161 y Fn(\))1340 1261 y(\()p Fm(E)23 b Fl(\000)c Fm(\020)6 b Fn(\))p Fm(\013)1667 1273 y Fi(k)1708 1261 y Fm(\015)1751 1273 y Fi(k)1810 1261 y Fn(+)18 b(2)p Fm(\014)1982 1273 y Fi(k)2023 1261 y Fm(\015)2066 1273 y Fi(k)2241 1261 y Fl(\000)p Fn(\(\()p Fm(E)24 b Fl(\000)18 b Fm(\020)6 b Fn(\))p Fm(\015)2660 1231 y Fk(2)2655 1285 y Fi(k)2716 1261 y Fn(+)18 b(2)p Fm(\015)2884 1273 y Fi(k)2924 1261 y Fm(\016)2961 1273 y Fi(k)3002 1261 y Fn(\))3127 1095 y Fh(\023)1066 1445 y Fn(=:)1237 1328 y Fh(\022)1340 1394 y Fm(a)1384 1406 y Fi(k)1507 1394 y Fl(\000)p Fm(b)1608 1406 y Fi(k)1344 1494 y Fm(b)1380 1506 y Fi(k)1536 1494 y Fm(d)1579 1506 y Fi(k)1690 1328 y Fh(\023)515 1677 y Fn(and)27 b(hence)h(w)n(e)f(ha)n(v)n(e)f(\(3\))i(also)f(for)g Fm(n)1698 1689 y Fi(k)1766 1677 y Fn(o)r(dd.)639 1777 y(No)n(w)f(assume)g(that)h(there)f(exists)h(a)f(deca)n(ying)f(solution) h Fm(\036)h Fn(of)g(\(2\).)36 b(Then)27 b Fl(k)p Fm(M)3127 1789 y Fi(E)3182 1777 y Fn(\()p Fm(n)3264 1789 y Fi(k)3305 1777 y Fn(\))p Fl(k)515 1876 y Fn(and)i(th)n(us)h(max)o Fl(fj)p Fm(a)1126 1888 y Fi(k)1166 1876 y Fl(j)p Fm(;)14 b Fl(j)p Fm(b)1285 1888 y Fi(k)1326 1876 y Fl(j)p Fm(;)g Fl(j)p Fm(d)1452 1888 y Fi(k)1493 1876 y Fl(jg)29 b Fn(tends)h(to)f (in\014nit)n(y)-7 b(.)43 b(By)30 b(assumption)f(\(ii\),)h(on)g(a)f (subse-)515 1976 y(quence)c Fm(n)837 1988 y Fi(k)872 1996 y Fd(j)933 1976 y Fn(w)n(e)g(ha)n(v)n(e)f Fl(j)p Fm(a)1309 1988 y Fi(k)1344 1996 y Fd(j)1394 1976 y Fn(+)14 b Fm(d)1516 1988 y Fi(k)1551 1996 y Fd(j)1587 1976 y Fl(j)23 b(\024)g Fm(C)1780 1988 y Fk(1)1817 1976 y Fn(.)36 b(Using)26 b(this)f(and)h(the)g(relation)e(det)14 b Fm(M)3080 1988 y Fi(E)3136 1976 y Fn(\()p Fm(n)3218 1988 y Fi(k)3259 1976 y Fn(\))23 b(=)515 2076 y Fm(a)559 2088 y Fi(k)599 2076 y Fm(d)642 2088 y Fi(k)702 2076 y Fn(+)18 b Fm(b)821 2045 y Fk(2)821 2099 y Fi(k)885 2076 y Fn(=)k(1,)27 b(w)n(e)h(get)1445 2275 y(min)q Fl(fj)p Fm(a)1693 2287 y Fi(k)1728 2295 y Fd(j)1763 2275 y Fl(j)p Fm(;)14 b Fl(j)p Fm(b)1882 2287 y Fi(k)1917 2295 y Fd(j)1952 2275 y Fl(j)p Fm(;)g Fl(j)p Fm(d)2078 2287 y Fi(k)2113 2295 y Fd(j)2149 2275 y Fl(jg)22 b(!)h(1)p Fm(:)825 b Fn(\(4\))515 2424 y(No)n(w,)31 b(with)g Fm(v)g Fn(=)c(\010\(0\),)32 b(w)n(e)e(ha)n(v)n(e)f Fl(k)p Fm(M)1749 2436 y Fi(E)1804 2424 y Fn(\()p Fm(n)1886 2436 y Fi(k)1927 2424 y Fn(\))p Fm(v)s Fl(k)f(!)g Fn(0)i(as)g Fm(k)h Fl(!)d(1)p Fn(.)46 b(W)-7 b(e)31 b(see)f(that,)h(as)f(an)515 2524 y(elemen)n(t)35 b(of)f Fm(P)12 b Fn(\()p Fj(R)1081 2494 y Fk(2)1125 2524 y Fn(\),)37 b Fm(v)h Fl(6)p Fn(=)c(\(1)p Fm(;)14 b Fn(0\))1579 2494 y Fi(T)1631 2524 y Fn(,)37 b(for)d(otherwise)g Fm(a)2246 2536 y Fi(k)2287 2524 y Fm(;)14 b(b)2360 2536 y Fi(k)2435 2524 y Fl(!)35 b Fn(0)f(and)h(hence)g Fm(d)3079 2536 y Fi(k)3155 2524 y Fl(!)g(1)p Fn(,)515 2624 y(con)n(tradicting)30 b(assumption)h(\(ii\).)49 b(Similarly)-7 b(,)32 b(it)g(follo)n(ws)e(that)i(in)f Fm(P)12 b Fn(\()p Fj(R)2821 2593 y Fk(2)2865 2624 y Fn(\),)33 b Fm(v)f Fl(6)p Fn(=)d(\(0)p Fm(;)14 b Fn(1\))3304 2593 y Fi(T)3356 2624 y Fn(.)515 2723 y(Therefore)26 b Fm(v)g Fn(=)d(\(1)p Fm(;)14 b(\030)t Fn(\))1228 2693 y Fi(T)1308 2723 y Fn(in)28 b Fm(P)12 b Fn(\()p Fj(R)1556 2693 y Fk(2)1599 2723 y Fn(\))28 b(with)g(some)f Fm(\030)h Fl(6)p Fn(=)22 b(0.)37 b(W)-7 b(e)28 b(get)1432 2906 y Fm(a)1476 2918 y Fi(k)1535 2906 y Fl(\000)18 b Fm(\030)t(b)1694 2918 y Fi(k)1758 2906 y Fl(!)23 b Fn(0)p Fm(;)37 b(b)2002 2918 y Fi(k)2061 2906 y Fn(+)18 b Fm(\030)t(d)2227 2918 y Fi(k)2291 2906 y Fl(!)23 b Fn(0)p Fm(:)515 3088 y Fn(Hence,)1676 3188 y Fm(a)1720 3200 y Fi(k)1779 3188 y Fn(+)18 b Fm(\030)1902 3154 y Fk(2)1940 3188 y Fm(d)1983 3200 y Fi(k)2047 3188 y Fl(!)23 b Fn(0)p Fm(:)515 3337 y Fn(In)j(particular,)g(on)g(the)h (subsequence)f Fm(n)1797 3349 y Fi(k)1832 3357 y Fd(j)1894 3337 y Fn(w)n(e)g(ha)n(v)n(e)g(b)r(oth)g Fm(a)2444 3349 y Fi(k)2479 3357 y Fd(j)2531 3337 y Fn(+)16 b Fm(\030)2652 3307 y Fk(2)2689 3337 y Fm(d)2732 3349 y Fi(k)2767 3357 y Fd(j)2826 3337 y Fl(!)23 b Fn(0)j(and)h Fl(j)p Fm(a)3228 3349 y Fi(k)3263 3357 y Fd(j)3314 3337 y Fn(+)515 3437 y Fm(d)558 3449 y Fi(k)593 3457 y Fd(j)628 3437 y Fl(j)37 b(\024)f Fm(C)848 3449 y Fk(1)885 3437 y Fn(,)i(so)d(that)g(\(4\))h (implies)g Fm(\030)1715 3407 y Fk(2)1788 3437 y Fn(=)g(1.)60 b(But)36 b(then)g(tr)p Fm(M)2530 3449 y Fi(E)2586 3437 y Fn(\()p Fm(n)2668 3449 y Fi(k)2709 3437 y Fn(\))g(=)g Fm(a)2922 3449 y Fi(k)2987 3437 y Fn(+)23 b Fm(d)3118 3449 y Fi(k)3195 3437 y Fl(!)36 b Fn(0,)515 3537 y(con)n(tradicting)26 b(assumption)h(\(iii\).)p 3323 3512 57 4 v 3323 3562 4 50 v 3376 3562 V 3323 3565 57 4 v 639 3686 a(The)h(ab)r(o)n(v)n(e)e (pro)r(of)h(sho)n(ws)g(in)g(fact)h(the)g(follo)n(wing:)515 3869 y Fg(Corollary)j(1)42 b Ff(Fix)26 b(some)h(ener)l(gy)g Fm(E)h Fl(2)23 b Fj(R)p Ff(.)44 b(Supp)l(ose)27 b(as)f(in)h(The)l(or)l (em)g(1)g(that)g(the)f(p)l(oten-)515 3968 y(tial)36 b(on)f(the)g(right) h(half-line)h(starts)e(with)h(in\014nitely)f(many)h(p)l(alindr)l(omes)h (and)e(supp)l(ose)515 4068 y(further)28 b(that)g(the)h(tr)l(ac)l(es)f (of)h(the)f(asso)l(ciate)l(d)i(tr)l(ansfer)e(matric)l(es)h(ar)l(e)f(b)l (ounde)l(d)h(on)f(a)h(sub-)515 4168 y(se)l(quenc)l(e.)59 b(Then)38 b(the)f(existenc)l(e)f(of)i(a)f(de)l(c)l(aying)h(solution)g (of)f(\(2\))g(implies)i(that)e(these)515 4267 y(tr)l(ac)l(es)26 b(c)l(onver)l(ge)h(to)g(zer)l(o)g(and)g(the)f(initial)i(c)l(ondition)g (of)f(the)g(de)l(c)l(aying)h(solution)f(is)g(e)l(qual)515 4367 y(to)i(either)i Fn(\(1)p Fm(;)14 b Fn(1\))1036 4337 y Fi(T)1118 4367 y Ff(or)30 b Fn(\(1)p Fm(;)14 b Fl(\000)p Fn(1\))1475 4337 y Fi(T)1556 4367 y Ff(in)30 b Fm(P)12 b Fn(\()p Fj(R)1809 4337 y Fk(2)1852 4367 y Fn(\))p Ff(.)639 4550 y Fn(This)36 b(result)f(can)h(b)r(e)g(in)n(terpreted)f(as)g(a)g (result)h(for)f(half-line)h(op)r(erators)d(on)j Fm(`)3218 4519 y Fk(2)3254 4550 y Fn(\()p Fj(N)5 b Fn(\))515 4649 y(where)38 b(one)h(has)g(to)g(imp)r(ose)g(a)f(b)r(oundary)h(condition)f (at)h(the)h(origin)e(to)h(ensure)f(self-)515 4749 y(adjoin)n(tness.)67 b(On)38 b(the)g(set)g(of)g(energies)f(where)h(the)g(transfer)f(matrix)g (traces)g(do)h(not)515 4848 y(div)n(erge)20 b(to)i(in\014nit)n(y)-7 b(,)23 b(one)e(therefore)g(has)g(absence)g(of)g(deca)n(ying)g(\(and)h (th)n(us)f Fm(`)2941 4818 y Fk(2)2978 4848 y Fn(-\))h(solutions)515 4948 y(for)27 b(all)g(but)h(t)n(w)n(o)f(b)r(oundary)g(conditions.)1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fo(3)134 b(Absence)125 b(of)f(Eigen)l(v)-7 b(alues)127 b(for)d(Substitution)716 672 y(Sc)l(hr\177)-67 b(odinger)45 b(Op)t(erators)515 854 y Fn(In)50 b(this)g(section,)k(w)n (e)c(w)n(an)n(t)f(to)g(apply)h(our)f(criterion)f(to)i(the)g(particular) e(case)h(of)515 954 y(Sc)n(hr\177)-42 b(odinger)44 b(op)r(erators)g (with)j(p)r(oten)n(tials)f(generated)f(b)n(y)h(primitiv)n(e)g (substitutions.)515 1054 y(Some)40 b(classes)f(of)h(suc)n(h)g(op)r (erators)e(ha)n(v)n(e)h(already)g(b)r(een)i(sho)n(wn)e(to)h(exhibit)h (absence)515 1153 y(of)31 b(eigen)n(v)-5 b(alues)29 b([6,)i(9)o(,)g(11) o(,)g(12)o(,)g(15)o(,)g(16)o(,)g(21)o(,)g(23)o(,)g(24)o(,)g(34)o(].)47 b(Here)30 b(w)n(e)h(will)g(sho)n(w)f(ho)n(w)g(the)515 1253 y(application)d(of)g(our)g(criterion)g(allo)n(ws)f(one)h(to)h (treat)f(an)g(additional)g(sub)r(class.)639 1352 y(Sc)n(hr\177)-42 b(odinger)30 b(op)r(erators)g(with)i(p)r(oten)n(tials)f(generated)g(b)n (y)g(primitiv)n(e)h(substitutions)515 1452 y(ha)n(v)n(e)d(b)r(een)h (studied)h(mainly)f(b)r(ecause)g(of)g(their)g(relev)-5 b(ance)30 b(to)g(quasicrystals)e(and)i(their)515 1552 y(exotic)j(sp)r(ectral)h(prop)r(erties.)55 b(Since)34 b(the)g(disco)n(v)n(ery)e(of)i(quasicrystals)e(b)n(y)h(Shec)n(h)n(tman) 515 1651 y(et)d(al.)g(in)g(1984)e([33)o(],)j(a)f(lot)g(of)g(e\013ort)f (has)h(b)r(een)g(made)g(to)g(\014nd)h(appropriate)d(structural)515 1751 y(mo)r(dels.)35 b(The)24 b(t)n(w)n(o)f(most)h(hea)n(vily)f (studied)h(mo)r(dels)g(are)f(generated)f(b)n(y)i(either)g(a)f(cut)h (and)515 1851 y(pro)5 b(ject)23 b(sc)n(heme)h(or)f(a)h(substitution)h (pro)r(cess.)35 b(On)24 b(the)g(other)g(hand,)h(there)f(has)f(b)r(een)i (in-)515 1950 y(tense)c(researc)n(h)e(activit)n(y)i(on)g(Sc)n(hr\177) -42 b(odinger)19 b(op)r(erators)h(with)h(purely)g(singular)f(con)n(tin) n(uous)515 2050 y(sp)r(ectra)30 b(o)n(v)n(er)f(the)i(last)g(decade)f (and)h(it)g(turned)g(out)g(that)g(as)f(a)h(rule,)g(one-dimensional)515 2149 y(Sc)n(hr\177)-42 b(odinger)31 b(op)r(erators)g(with)i(p)r(oten)n (tials)g(generated)f(b)n(y)g(primitiv)n(e)h(substitution)h(ap-)515 2249 y(paren)n(tly)d(exhibit)h(this)g(\\exotic")e(sp)r(ectral)i(t)n(yp) r(e.)49 b(While)33 b(absence)e(of)h(absolutely)f(con-)515 2349 y(tin)n(uous)23 b(sp)r(ectrum)g(follo)n(ws)f(in)h(full)h (generalit)n(y)d(from)i(Kotani)f([26)o(])i(and)f(Last)f(and)h(Simon)515 2448 y([27)o(],)k(absence)f(of)g(p)r(oin)n(t)h(sp)r(ectrum)f(is)h(not)f (kno)n(wn)g(in)h(similar)e(generalit)n(y)-7 b(.)36 b(Ho)n(w)n(ev)n(er,) 24 b(no)515 2548 y(coun)n(terexample)d(is)i(kno)n(wn,)h(and)e(there)h (is)g(a)g(h)n(uge)f(n)n(um)n(b)r(er)h(of)g(p)r(ositiv)n(e)f(partial)h (results;)515 2648 y(see)36 b([13)o(])h(for)f(a)g(review)g(and)g([9])h (for)f(the)h(conjecture)f(that)h(the)g(h)n(yp)r(othesis)f(su\016cien)n (t)515 2747 y(to)29 b(pro)n(v)n(e)f(the)h(singularit)n(y)f(of)i(the)g (sp)r(ectrum,)g(that)f(is,)h(semi-primitivit)n(y)f(of)g(a)g(reduced)515 2847 y(trace)d(map)h(and)g(existence)f(of)h(a)g(square)e(in)j Fm(u)e Fn(\(Theorem)h(2)f(b)r(elo)n(w\),)h(are)f(also)g(su\016cien)n(t) 515 2946 y(to)f(pro)n(v)n(e)f(absence)i(of)f(eigen)n(v)-5 b(alues)25 b(.)36 b(Apart)26 b(from)f(results)g(on)h(the)g(sp)r(ectral) f(t)n(yp)r(e,)i(other)515 3046 y(in)n(teresting)d(results)g(ha)n(v)n(e) f(b)r(een)j(obtained)e(for)g(these)h(op)r(erators,)e(suc)n(h)i(as)f (zero-measure)515 3146 y(Can)n(tor)k(sp)r(ectrum)h([6)o(,)h(8)o(,)f(9,) g(35)o(],)h(general)e(gap-lab)r(elling)f(\(heuristic)j([28)o(],)f Fm(K)6 b Fn(-theoretic)515 3245 y([7)o(])34 b(and)f(constructiv)n(e)f ([30)o(]\),)j(and)e(results)f(on)h(the)h(op)r(ening)f(of)g(gaps)f(at)h (lo)n(w)g(coupling)515 3345 y(\(heuristic)27 b([28)o(])h(and)g(precise) e([5,)i(6)o(]\))g(and)g(large)e(coupling)h([30)o(].)515 3577 y Fc(3.1)112 b(A)37 b(class)g(of)h(mo)s(dels)f(with)f(empt)m(y)h (p)s(oin)m(t)f(sp)s(ectrum)515 3731 y Fn(Let)27 b(us)h(\014rst)f (recall)g(some)g(de\014nitions)h(concerning)e(these)i(op)r(erators.)639 3830 y(A)37 b Ff(substitution)e Fn([29)o(])i(is)f(a)g(map)g Fm(S)41 b Fn(from)36 b(a)g(\014nite)g(alphab)r(et)h Fl(A)f Fn(to)g(the)h(set)f Fl(A)3237 3800 y Fe(\003)3312 3830 y Fn(of)515 3930 y(w)n(ords)e(on)i Fl(A)p Fn(,)j(whic)n(h)d(can)g(b)r (e)g(naturally)f(extended)i(to)f(a)f(map)h(from)g Fl(A)2920 3900 y Fe(\003)2995 3930 y Fn(to)g Fl(A)3171 3900 y Fe(\003)3245 3930 y Fn(and)515 4030 y(also)31 b(to)h(a)g(map)g(from)f Fl(A)1321 3999 y Fb(N)1400 4030 y Fn(to)h Fl(A)1572 3999 y Fb(N)1618 4030 y Fn(.)51 b Fm(S)37 b Fn(is)32 b(said)g(to)g(b)r(e)g Ff(primitive)i Fn(if)f(there)f(exists)g Fm(k)h Fl(2)e Fj(N)515 4129 y Fn(suc)n(h)26 b(that)i(for)e(ev)n(ery)g(pair)g(\()p Fm(\013;)14 b(\014)t Fn(\))24 b Fl(2)g(A)1770 4099 y Fk(2)1807 4129 y Fn(,)j Fm(S)1913 4099 y Fi(k)1954 4129 y Fn(\()p Fm(\013)p Fn(\))h(con)n(tains)e Fm(\014)t Fn(.)37 b(A)27 b Ff(substitution)h(se)l(quenc)l(e)515 4229 y Fm(u)38 b Fn(is)g(a)g(\014xed)h(p)r(oin)n(t)g(of)f Fm(S)44 b Fn(giv)n(en)37 b(b)n(y)i(inde\014nite)g(iteration)f(of)g Fm(S)44 b Fn(on)38 b(a)g(letter)h Fm(a)i Fl(2)h(A)515 4328 y Fn(suc)n(h)31 b(that)h Fm(S)5 b Fn(\()p Fm(a)p Fn(\))32 b(b)r(egins)g(with)g Fm(a)p Fn(.)50 b(The)31 b Ff(hul)t(l)i Fn(of)f Fm(u)p Fn(,)g(\012)2259 4340 y Fi(u)2303 4328 y Fn(,)h(is)e(de\014ned)h(as)f(the)i(set)e(of)h(t)n(w)n (o-)515 4428 y(sided)e(in\014nite)g(sequences)f(o)n(v)n(er)f Fl(A)j Fn(that)f(ha)n(v)n(e)e(all)i(their)g(\014nite)g(sub)n(w)n(ords)e (o)r(ccurring)h(in)515 4528 y Fm(u)p Fn(.)62 b(Fix)36 b(some)f(function)i Fm(f)46 b Fn(:)37 b Fl(A)h(!)f Fj(R)p Fn(.)68 b(One)36 b(sa)n(ys)f(that)h(a)g(Sc)n(hr\177)-42 b(odinger)34 b(op)r(erator)g(of)515 4627 y(t)n(yp)r(e)j(\(1\))h(is)g Ff(asso)l(ciate)l(d)i(with)f Fm(S)44 b Ff(and)39 b Fm(f)46 b Fn(if)38 b(the)g(sequence)f(\()p Fm(V)19 b Fn(\()p Fm(n)p Fn(\)\))2725 4639 y Fi(n)p Fe(2)p Fa(Z)2900 4627 y Fn(has)37 b(the)g(form)515 4727 y Fm(V)19 b Fn(\()p Fm(n)p Fn(\))34 b(=)f Fm(f)9 b Fn(\()p Fm(!)962 4739 y Fi(n)1007 4727 y Fn(\))35 b(for)e(some)h Fm(!)i Fl(2)e Fn(\012)1659 4739 y Fi(u)1703 4727 y Fn(.)56 b(It)34 b(follo)n(ws)g(from)f(general)g(principles)h(\(see,)h(e.g.,)515 4827 y([13)o(]\))27 b(that)g(primitivit)n(y)f(of)h Fm(S)k Fn(implies)c(the)g(existence)f(of)h(a)f(closed)g(set)g(\006)d Fl(\022)g Fj(R)33 b Fn(suc)n(h)26 b(that)515 4926 y(the)g(sp)r(ectrum)f (of)h(ev)n(ery)e(asso)r(ciated)g(op)r(erator)g Fm(H)32 b Fn(is)25 b(equal)g(to)h(\006.)36 b(Ho)n(w)n(ev)n(er,)24 b(in)h(general)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fn(the)28 b(sp)r(ectral)f(t)n(yp)r(e)h(of)f Fm(H)35 b Fn(need)27 b(not)h(b)r(e)g(indep)r(enden)n(t)g(of)g Fm(!)s Fn(.)639 623 y(W)-7 b(e)41 b(will)f(alw)n(a)n(ys)e(assume)h(in)i (the)f(follo)n(wing)f(that)h Fm(S)45 b Fn(and)40 b Fm(f)49 b Fn(are)39 b(suc)n(h)g(that)i(the)515 722 y(p)r(oten)n(tials)33 b Fm(V)52 b Fn(are)32 b(not)i(p)r(erio)r(dic)f(since)g(the)h(sp)r (ectral)f(theory)f(of)i(the)f(p)r(erio)r(dic)h(case)e(is)515 822 y(w)n(ell)27 b(established.)639 922 y(Let)k(us)g(recall)e(some)h (notions)h(whic)n(h)f(are)g(crucial)g(to)g(the)h(approac)n(h)e(of)i([9) o(].)47 b(De\014ne)515 1021 y(for)27 b(an)n(y)g(w)n(ord)f Fm(w)g Fn(=)c Fm(w)1235 1033 y Fk(1)1287 1021 y Fm(:)14 b(:)g(:)g(w)1457 1033 y Fi(m)1543 1021 y Fl(2)24 b(A)1688 991 y Fe(\003)1754 1021 y Fn(and)j(ev)n(ery)g Fm(E)h Fl(2)23 b Fj(R)p Fn(,)1226 1201 y Fm(M)1307 1213 y Fi(E)1363 1201 y Fn(\()p Fm(w)r Fn(\))h(=)f Fm(T)1649 1213 y Fi(E)1704 1201 y Fn(\()p Fm(f)9 b Fn(\()p Fm(w)1877 1213 y Fi(n)1923 1201 y Fn(\)\))19 b Fl(\002)f(\001)c(\001)g(\001)19 b(\002)f Fm(T)2337 1213 y Fi(E)2392 1201 y Fn(\()p Fm(f)9 b Fn(\()p Fm(w)2565 1213 y Fk(1)2603 1201 y Fn(\)\))515 1380 y(and)1514 1480 y Fm(M)1604 1437 y Fk(\()p Fi(k)q Fk(\))1595 1504 y Fi(E)1696 1480 y Fn(\()p Fm(w)r Fn(\))24 b(=)f Fm(M)2014 1492 y Fi(E)2070 1480 y Fn(\()p Fm(S)2158 1446 y Fi(k)2198 1480 y Fn(\()p Fm(w)r Fn(\)\))p Fm(:)515 1627 y Fn(The)f(substitution)h (rule)e(naturally)h(leads)f(to)h(recursiv)n(e)e(relations)h(b)r(et)n(w) n(een)h(the)h(matrices)515 1739 y Fm(M)605 1695 y Fk(\()p Fi(k)q Fk(\))596 1763 y Fi(E)697 1739 y Fn(\()p Fm(w)r Fn(\).)43 b(F)-7 b(rom)28 b(these)i(recursions)d(one)i(can)f(obtain)h (an)g(ev)n(en)f(more)h(useful)g(system)g(of)515 1838 y(recursiv)n(e)j(equations)h(for)h(the)g(traces)f(of)h(these)g (matrices.)55 b(In)34 b(general)f(there)h(exists)f(a)515 1938 y(\014nite)g(subset)f(of)h(w)n(ords)e Fl(B)j(\032)c(A)1584 1908 y Fe(\003)1656 1938 y Fn(con)n(taining)h Fl(A)i Fn(for)f(whic)n(h)h(these)f(equations)g(yield)g(a)515 2037 y(closed)e(set)h(of)g(recursiv)n(e)e(p)r(olynomial)h(equations,)h (whic)n(h)g(is)g(called)g(the)g Ff(tr)l(ac)l(e)i(map)p Fn(.)48 b(It)515 2137 y(turns)28 b(out)g(that)g(to)g(eac)n(h)f(trace)g (map)h(one)f(can)h(asso)r(ciate)f(a)g Ff(r)l(e)l(duc)l(e)l(d)j(tr)l(ac) l(e)g(map)f Fn(that)f(is)515 2237 y Ff(monomial)p Fn(.)40 b(T)-7 b(o)28 b(this)h(reduced)e(trace)h(map,)g(one)g(can)g(then)h (asso)r(ciate)e(a)g(substitution)3337 2216 y(^)3323 2237 y Fm(S)515 2336 y Fn(on)h Fl(B)i Fn(whose)d(prop)r(erties)h(are)f (ultimately)h(crucial)g(for)f(the)i(sp)r(ectral)e(analysis.)37 b(In)29 b(short)515 2436 y(terms,)e(the)g(reduced)g(trace)f(map)h(is)h (obtained)e(b)n(y)h(k)n(eeping)g(in)g(the)h(recursiv)n(e)d(equations) 515 2536 y(the)31 b(terms)g(of)g(highest)g(degree)f(whic)n(h)h (determine)g(the)h(b)r(eha)n(vior)e(of)h(the)g(norms)g(of)g(the)515 2635 y(transfer)26 b(matrices)h(at)g(large)f Fm(n)p Fn(,)h(allo)n(wing) f(one,)h(under)h(the)g(h)n(yp)r(othesis)e(of)i(the)g(theorem)515 2735 y(b)r(elo)n(w,)38 b(to)e(iden)n(tify)h(the)f(sp)r(ectrum)g(with)h (the)g(set)f(of)g(energies)f(with)i(zero)e(Ly)n(apuno)n(v)515 2834 y(exp)r(onen)n(t)e(and)h(th)n(us)g(to)f(apply)h(Kotani's)f (theorem)g([26)o(])h(to)g(pro)n(v)n(e)e(the)i(singularit)n(y)e(of)515 2934 y(the)k(sp)r(ectrum;)41 b(see)35 b([9])h(for)g(details)f(of)h(the) h(pro)r(of.)61 b(W)-7 b(e)37 b(call)e(suc)n(h)h(a)g(substitution)3337 2913 y(^)3323 2934 y Fm(S)515 3034 y Ff(semi-primitive)29 b Fn(if:)593 3213 y(\(i\))42 b(There)25 b(exists)g Fl(C)j(\032)23 b(B)k Fn(suc)n(h)e(that)1806 3192 y(^)1792 3213 y Fm(S)5 b Fl(j)1871 3225 y Fe(C)1940 3213 y Fn(is)25 b(a)g(primitiv)n(e)g (substitution)i(from)e Fl(C)30 b Fn(to)25 b Fl(C)3318 3183 y Fe(\003)3356 3213 y Fn(.)570 3378 y(\(ii\))42 b(There)30 b(exists)f Fm(k)k Fn(suc)n(h)c(that)h(for)f(all)h Fm(\014)h Fl(2)c(B)s Fn(,)2175 3357 y(^)2161 3378 y Fm(S)2217 3348 y Fi(k)2257 3378 y Fn(\()p Fm(\014)t Fn(\))k(con)n(tains)e(at)g (least)h(one)f(letter)722 3478 y(from)f Fl(C)5 b Fn(.)639 3657 y(The)28 b(follo)n(wing)e(theorem)i(w)n(as)e(pro)n(v)n(en)g(in)i ([9]:)515 3837 y Fg(Theorem)i(2)41 b Ff(Assume)28 b(that)g Fm(S)k Ff(is)d(primitive)h(and)e(has)h(a)f(\014xe)l(d)g(p)l(oint)g Fm(u)p Ff(,)h Fm(f)36 b Ff(is)29 b(such)f(that)515 3936 y(the)c(asso)l(ciate)l(d)h(p)l(otentials)g(ar)l(e)f(ap)l(erio)l(dic,)29 b(and)24 b(the)g(fol)t(lowing)j(c)l(onditions)e(ar)l(e)f(satis\014e)l (d:)593 4100 y Fn(\(i\))42 b Fm(u)30 b Ff(c)l(ontains)f(the)h(squar)l (e)g(of)h(a)f(wor)l(d)g(in)g Fl(B)s Ff(.)570 4265 y Fn(\(ii\))42 b Ff(Ther)l(e)24 b(exists)f(a)h(tr)l(ac)l(e)f(map)h(whose)g(asso)l (ciate)l(d)h(substitution)2685 4244 y Fn(^)2671 4265 y Fm(S)j Ff(is)c(semi-primitive.)515 4428 y(Then)f(for)g(every)g Fm(!)j Fl(2)d Fn(\012)1277 4440 y Fi(u)1321 4428 y Ff(,)h(the)e(sp)l(e) l(ctrum)g(of)h(the)g(op)l(er)l(ator)g Fm(H)29 b Ff(in)23 b Fn(\(1\))f Ff(with)h Fm(V)c Fn(\()p Fm(n)p Fn(\))k(=)g Fm(f)9 b Fn(\()p Fm(!)3302 4440 y Fi(n)3347 4428 y Fn(\))515 4528 y Ff(is)30 b(singular)g(and)g(supp)l(orte)l(d)h(on)e(a)h(set)g(of) g(zer)l(o)h(L)l(eb)l(esgue)e(me)l(asur)l(e.)639 4707 y Fn(Of)f(sp)r(ecial)f(in)n(terest)g(for)g(an)g(application)f(of)i(our) e(criterion)h(from)g(Section)g(2)g(to)g(sub-)515 4807 y(stitution)32 b(mo)r(dels)f(is)g(the)g(fact)h(that)f(semi-primitivit)n (y)g(of)g(the)h(trace)e(map)h(implies)h(the)515 4907 y(existence)37 b(of)g(non-div)n(ergen)n(t)f(subsequences)g(of)h(trace)g (map)g(iterates)g(for)g(energies)f(in)515 5006 y(the)f(sp)r(ectrum)g ([9)o(,)i(Lemma)d(3.4].)57 b(It)35 b(has)f(b)r(een)h(sho)n(wn)f(in)h ([9])g(that)f(semi-primitivit)n(y)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fn(of)35 b(the)h(deriv)n(ed)e(substitution)1550 502 y(^)1536 523 y Fm(S)40 b Fn(holds)35 b(for)g(man)n(y)f(prominen)n (t)h(substitutions)h(includ-)515 623 y(ing)h(\(in)h(the)g(case)e(where) h Fl(A)j Fn(=)f Fl(f)p Fm(a;)14 b(b)p Fl(g)p Fn(\))36 b(Fib)r(onacci)h(\()p Fm(a)j Fl(7!)f Fm(ab;)14 b(b)39 b Fl(7!)g Fm(a)p Fn(\),)h(p)r(erio)r(d)e(dou-)515 722 y(bling)31 b(\()p Fm(a)d Fl(7!)h Fm(ab;)14 b(b)28 b Fl(7!)h Fm(aa)p Fn(\),)i(binary)g(non-Pisot)e(\()p Fm(a)g Fl(7!)g Fm(ab;)14 b(b)27 b Fl(7!)i Fm(aaa)p Fn(\),)j(and)f(Th)n(ue-Morse)515 822 y(\()p Fm(a)23 b Fl(7!)g Fm(ab;)14 b(b)22 b Fl(7!)h Fm(ba)p Fn(,)k(to)h(b)r(e)g(discussed)f(in)h(detail)g(b)r(elo)n(w\).) 639 938 y(Let)38 b Fm(x)845 950 y Fi(n)890 938 y Fn(\()p Fm(E)5 b Fn(\))41 b(=)d(tr\()p Fm(M)1351 895 y Fk(\()p Fi(n)p Fk(\))1342 962 y Fi(E)1448 938 y Fn(\()p Fm(a)p Fn(\)\),)j(where)c Fm(a)g Fn(is)h(the)f(\014rst)g(sym)n(b)r(ol)g(of)h Fm(u)p Fn(.)66 b(As)37 b(a)g(conse-)515 1049 y(quence)e(w)n(e)g(get)h (that)g(in)f(the)h(case)f(of)h(semi-primitiv)n(e)2371 1028 y(^)2357 1049 y Fm(S)t Fn(,)i(for)d(ev)n(ery)f(energy)h Fm(E)40 b Fn(from)515 1149 y(the)27 b(sp)r(ectrum,)h Fl(j)p Fm(x)1113 1161 y Fi(n)1158 1149 y Fn(\()p Fm(E)5 b Fn(\))p Fl(j)28 b Fn(is)f(b)r(ounded)h(on)e(a)h(subsequence)g(\(whic) n(h)g(ma)n(y)f(dep)r(end)i(on)f(the)515 1248 y(energy\).)639 1348 y(F)-7 b(or)25 b(concrete)f(mo)r(dels)i(with)g(a)e(semi-primitiv)n (e)2199 1327 y(^)2185 1348 y Fm(S)30 b Fn(and)25 b(whic)n(h)h(displa)n (y)e(the)i(required)515 1448 y(palindromic)i(symmetries,)g(w)n(e)g(can) g(therefore)g(fo)r(cus)h(our)e(atten)n(tion)i(on)f Ff(lower)i Fn(b)r(ounds)515 1547 y(for)d(a)g(subsequence)g(of)h Fl(j)p Fm(x)1345 1559 y Fi(n)1390 1547 y Fn(\()p Fm(E)5 b Fn(\))p Fl(j)p Fn(.)639 1647 y(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h (follo)n(wing)e(theorem:)515 1829 y Fg(Theorem)k(3)41 b Ff(Assume)28 b(that)g Fm(S)k Ff(is)d(primitive)h(and)e(has)h(a)f (\014xe)l(d)g(p)l(oint)g Fm(u)p Ff(,)h Fm(f)36 b Ff(is)29 b(such)f(that)515 1929 y(the)c(asso)l(ciate)l(d)h(p)l(otentials)g(ar)l (e)f(ap)l(erio)l(dic,)29 b(and)24 b(the)g(fol)t(lowing)j(c)l(onditions) e(ar)l(e)f(satis\014e)l(d:)593 2112 y Fn(\(i\))42 b Fm(S)778 2082 y Fi(n)823 2112 y Fn(\()p Fm(a)p Fn(\))30 b Ff(is)g(a)h(p)l (alindr)l(ome)g(for)g(every)f Fm(n)p Ff(,)g(wher)l(e)h Fm(a)e Ff(is)h(the)g(\014rst)f(symb)l(ol)i(of)f Fm(u)p Ff(.)570 2278 y Fn(\(ii\))42 b Ff(Ther)l(e)24 b(exists)f(a)h(tr)l(ac)l (e)f(map)h(whose)g(asso)l(ciate)l(d)h(substitution)2685 2257 y Fn(^)2671 2278 y Fm(S)j Ff(is)c(semi-primitive.)547 2444 y Fn(\(iii\))42 b Ff(F)-6 b(or)30 b(every)h Fm(E)d Fl(2)23 b Fn(\006)p Ff(,)31 b Fm(x)1427 2456 y Fi(n)1472 2444 y Fn(\()p Fm(E)5 b Fn(\))24 b Fl(6!)f Fn(0)29 b Ff(as)h Fm(n)23 b Fl(!)g(1)p Ff(.)515 2626 y(Then)32 b(ther)l(e)f(is)h Fm(!)d Fl(2)d Fn(\012)1254 2638 y Fi(u)1328 2626 y Ff(such)32 b(that)f Fm(u)g Ff(is)h(the)f(r)l(estriction)h(of)g Fm(!)i Ff(to)d Fj(N)42 b Ff(and)31 b(the)h(op)l(er)l(ator)515 2726 y Fm(H)k Ff(in)30 b(\(1\))g(with)g Fm(V)19 b Fn(\()p Fm(n)p Fn(\))24 b(=)e Fm(f)9 b Fn(\()p Fm(!)1468 2738 y Fi(n)1513 2726 y Fn(\))30 b Ff(has)h(no)e(eigenvalues.)515 2909 y(Pr)l(o)l(of.)48 b Fn(W)-7 b(e)31 b(only)g(ha)n(v)n(e)f(to)g (construct)h(an)f(elemen)n(t)h Fm(!)h Fl(2)c Fn(\012)2435 2921 y Fi(u)2510 2909 y Fn(whose)i(restriction)g(to)h(the)515 3008 y(righ)n(t)22 b(half-line)h(coincides)g(with)g Fm(u)p Fn(.)35 b(The)24 b(assertion)d(is)i(then)h(a)f(consequence)f(of)h (Theorem)515 3108 y(1.)44 b(The)30 b(existence)g(of)g(suc)n(h)g(an)g (elemen)n(t)h(follo)n(ws)e(from)h(the)g(rep)r(etitivit)n(y)h(of)f Fm(u)g Fn(\(i.e.,)h(its)515 3208 y(\014nite)25 b(sub)n(w)n(ords)f(o)r (ccur)g(in\014nitely)i(often\))f(and)g(compactness)f(of)h(\012)2678 3220 y Fi(u)2747 3208 y Fn(\(since)g(it)g(is)g(clearly)515 3307 y(a)c(closed)g(subset)g(of)g(the)h(compact)f Fl(A)1682 3277 y Fb(Z)1725 3307 y Fn(\).)36 b(Namely)-7 b(,)22 b(using)f(these)h(t)n(w)n(o)f(prop)r(erties,)g(one)g(can)515 3407 y(construct)k(a)h(subsequence)g(of)g(elemen)n(ts)g(of)g(\012)g (whic)n(h)h(on)e(the)i(righ)n(t)f(half-line)g(con)n(v)n(erges)515 3506 y(p)r(oin)n(t)n(wise)f(to)h Fm(u)g Fn(and)g(then)g(c)n(ho)r(ose)f (a)h(con)n(v)n(erging)d(subsequence)j(b)n(y)g(compactness.)35 b(The)515 3606 y(limit)i Fm(!)j Fn(of)d(this)g(subsequence)f(then)i (coincides)e(with)h Fm(u)g Fn(on)f(the)i(righ)n(t)e(half-line.)64 b(By)515 3706 y(construction,)36 b(w)n(e)e(ha)n(v)n(e)g(that)h (assumptions)g(\(i\))g(and)g(\(iii\))h(imply)f(conditions)g(\(i\))g (and)515 3805 y(\(iii\))41 b(of)g(Theorem)e(1,)44 b(resp)r(ectiv)n(ely) -7 b(,)43 b(while,)h(as)c(men)n(tioned)h(ab)r(o)n(v)n(e,)h(assumption)e (\(ii\))515 3905 y(implies)28 b(condition)f(\(ii\))h(of)g(Theorem)f(1)g (b)n(y)g([9,)g(Lemma)h(3.4].)p 3323 3880 57 4 v 3323 3930 4 50 v 3376 3930 V 3323 3933 57 4 v 639 4054 a(If)23 b Fm(u)e Fn(con)n(tains)g(a)g(square,)h(whic)n(h)g(is)f(the)i(case,)f (for)f(example,)i(if)f(there)g(is)f(a)h(palindrome)515 4154 y(of)j(ev)n(en)g(length,)h(the)f(op)r(erator)f Fm(H)32 b Fn(ab)r(o)n(v)n(e)24 b(v)n(eri\014es)g(all)h(the)g(assumptions)g(of)g (Theorem)f(2.)515 4254 y(W)-7 b(e)28 b(can)f(therefore)g(state)g(the)h (follo)n(wing)f(corollary)e(to)i(Theorem)g(3.)515 4436 y Fg(Corollary)k(2)42 b Ff(Under)29 b(the)h(assumptions)g(of)g(The)l (or)l(em)h(3,)f(if)g Fm(u)f Ff(c)l(ontains)h(the)g(squar)l(e)f(of)515 4536 y(a)38 b(wor)l(d,)k(the)c(sp)l(e)l(ctrum)g(of)h Fm(H)45 b Ff(is)38 b(pur)l(ely)h(singular)f(c)l(ontinuous)g(and)h(supp) l(orte)l(d)f(on)h(a)515 4636 y(Cantor)30 b(set)f(of)i(L)l(eb)l(esgue)f (me)l(asur)l(e)f(zer)l(o.)515 4818 y(R)l(emark.)36 b Fn(This)25 b(corollary)d(is)j(a)g(partial)f(answ)n(er)f(to)i(the)g (conjecture)f(in)h([9])g(men)n(tioned)g(in)515 4918 y(the)j(in)n(tro)r (duction)f(of)h(this)f(section.)1926 5255 y(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fc(3.2)112 b(The)38 b(Th)m(ue-Morse)g(case)515 676 y Fn(As)31 b(a)g(\014rst)g(example,)g(w)n(e)g(consider)f(the)i(Th)n (ue-Morse)d(sequence;)k(compare,)e(in)g(partic-)515 776 y(ular,)37 b([5].)63 b(Generic)36 b(absence)f(of)h(eigen)n(v)-5 b(alues)35 b(w)n(as)g(sho)n(wn)h(b)n(y)g(Dely)n(on-P)n(eyri)n(\022)-39 b(ere)32 b([21)o(],)515 876 y(in)j([6)o(])g(\(in)h(an)e(implicit)i(w)n (a)n(y)d(that)j(will)f(b)r(e)g(made)f(explicit)i(here\),)g(and)f(b)n(y) f(Hof)h(et)g(al.)515 975 y([24)o(].)h(In)26 b(fact,)g(as)f(w)n(as)f (claimed)h(in)h([6])f(\(see)h(the)g(remark)e(at)h(the)h(end)g(of)f (Section)h(I)r(I\),)g(the)515 1075 y(palindromicit)n(y)g(of)h(some)f Fm(S)1425 1045 y Fi(n)1470 1075 y Fn(\()p Fm(a)p Fn(\))i(is)f(a)f (crucial)g(ingredien)n(t)h(in)g(the)h(pro)r(of.)36 b(Ho)n(w)n(ev)n(er,) 25 b(our)515 1174 y(result)31 b(additionally)g(yields)g(the)h(absence)f (of)g(deca)n(ying)g(solutions)f(at)i(+)p Fl(1)f Fn(and,)h(if)g(one)515 1274 y(considers)20 b(the)i(symmetric)f(extension)g(of)g Fm(u)g Fn(\(the)h(reader)e(ma)n(y)h(v)n(erify)f(that)i(this)g(sequence) 515 1374 y(b)r(elongs)h(to)h(\012)967 1386 y Fi(u)1010 1374 y Fn(\),)h(w)n(e)f(ev)n(en)f(ha)n(v)n(e)g(an)g(explicit)i(p)r (oten)n(tial)e(from)h(the)g(Th)n(ue-Morse)f(h)n(ull)h(for)515 1473 y(whic)n(h)29 b(ev)n(ery)g(solution)g(to)h(\(2\))g(with)g Fm(E)35 b Fn(in)30 b(the)g(sp)r(ectrum)g(deca)n(ys)f(neither)h(at)f(+)p Fl(1)h Fn(nor)515 1573 y(at)d Fl(\0001)p Fn(.)639 1673 y(Let)i(us)f(recall)f(the)i(de\014nition)g(of)f(the)h(Th)n(ue-Morse)e (substitution:)39 b(It)28 b(is)h(de\014ned)f(on)515 1772 y(the)e(alphab)r(et)g Fl(A)d Fn(=)g Fl(f)p Fm(a;)14 b(b)p Fl(g)24 b Fn(b)n(y)h Fm(S)5 b Fn(\()p Fm(a)p Fn(\))23 b(=)g Fm(ab)i Fn(and)h Fm(S)5 b Fn(\()p Fm(b)p Fn(\))23 b(=)g Fm(ba)p Fn(.)35 b(It)27 b(is)e(clearly)g(primitiv)n(e)g(and)515 1872 y(the)j(sequence)f Fm(u)22 b Fn(=)h(lim)1275 1884 y Fi(n)p Fe(!1)1467 1872 y Fm(S)1523 1842 y Fi(n)1568 1872 y Fn(\()p Fm(a)p Fn(\))28 b(is)f(called)g(the)h Ff(Thue-Morse)j(se)l(quenc)l(e)p Fn(.)639 1971 y(Since)22 b(one)f(also)f(has)h Fm(u)i Fn(=)f(lim)1572 1983 y Fi(n)p Fe(!1)1763 1971 y Fm(S)1819 1941 y Fk(2)p Fi(n)1897 1971 y Fn(\()p Fm(a)p Fn(\),)h Fm(S)2107 1941 y Fk(2)2166 1971 y Fn(generates)d(the)h(same)g(h)n(ull)h(and)f(hence)515 2071 y(the)33 b(same)g(family)g(of)h(asso)r(ciated)d(op)r(erators.)52 b(It)34 b(is)f(therefore)f(su\016cien)n(t)i(to)f(v)n(erify)f(all)515 2171 y(the)c(assumptions)f(of)g(Theorem)g(3)g(for)g(the)h(substitution) g Fm(S)2432 2141 y Fk(2)2469 2171 y Fn(.)639 2270 y(Ap)r(erio)r(dicit)n (y)23 b(of)h(the)f(asso)r(ciated)f(p)r(oten)n(tials)h(holds)f(for)h (all)g(non-constan)n(t)f(functions)515 2370 y Fm(f)9 b Fn(.)36 b(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n(e)639 2470 y(\(i\))d(The)f(iterates)f(of)h Fm(S)1353 2439 y Fk(2)1413 2470 y Fn(on)f Fm(a)h Fn(are)f(palindromes)g(of)h(ev)n(en)f (length)h(since)g Fm(S)2964 2439 y Fk(2)3001 2470 y Fn(\()p Fm(a)p Fn(\))g(=)g Fm(abba)515 2569 y Fn(and)k Fm(S)732 2539 y Fk(2)769 2569 y Fn(\()p Fm(b)p Fn(\))c(=)g Fm(baab)p Fn(.)639 2685 y(\(ii\))32 b(De\014ne)g Fm(y)1083 2697 y Fi(n)1128 2685 y Fn(\()p Fm(E)5 b Fn(\))30 b(=)f(tr\()p Fm(M)1569 2642 y Fk(\()p Fi(n)p Fk(\))1560 2709 y Fi(E)1666 2685 y Fn(\()p Fm(b)p Fn(\)\))p Fm(;)14 b(z)1874 2697 y Fi(n)1919 2685 y Fn(\()p Fm(E)5 b Fn(\))31 b(=)e(tr)o(\()p Fm(M)2360 2642 y Fk(\()p Fi(n)p Fk(\))2351 2709 y Fi(E)2457 2685 y Fn(\()p Fm(ab)p Fn(\)\).)49 b(The)32 b(trace)e(map)i(for)515 2785 y Fm(S)571 2754 y Fk(2)635 2785 y Fn(is)c(as)f(follo)n(ws)f(\(w)n (e)i(drop)f Fm(E)33 b Fn(for)27 b(simplicit)n(y)g(of)h(notation\):)1461 2984 y Fm(x)1508 2996 y Fi(n)p Fk(+1)1660 2984 y Fn(=)23 b Fm(x)1795 2996 y Fi(n)1841 2984 y Fm(y)1882 2996 y Fi(n)1926 2984 y Fm(z)1965 2996 y Fi(n)2029 2984 y Fl(\000)18 b Fm(x)2159 2950 y Fk(2)2159 3004 y Fi(n)2204 2984 y Fm(y)2248 2950 y Fk(2)2245 3004 y Fi(n)2309 2984 y Fn(+)g(2)1464 3122 y Fm(y)1505 3134 y Fi(n)p Fk(+1)1657 3122 y Fn(=)23 b Fm(x)1792 3134 y Fi(n)1837 3122 y Fm(y)1878 3134 y Fi(n)1923 3122 y Fm(z)1962 3134 y Fi(n)2025 3122 y Fl(\000)18 b Fm(x)2155 3088 y Fk(2)2155 3143 y Fi(n)2201 3122 y Fm(y)2245 3088 y Fk(2)2242 3143 y Fi(n)2305 3122 y Fn(+)g(2)1293 3261 y Fm(z)1332 3273 y Fi(n)p Fk(+1)1484 3261 y Fn(=)k Fm(x)1618 3226 y Fk(3)1618 3281 y Fi(n)1664 3261 y Fm(y)1708 3226 y Fk(3)1705 3281 y Fi(n)1750 3261 y Fm(z)1789 3273 y Fi(n)1852 3261 y Fn(+)46 b(lo)n(w)n(er)26 b(order)g(terms)639 3399 y(whic)n(h)i(giv)n(es)e(rise)h(to)h(a)f(semi-primitiv)n(e)g (reduced)g(trace)g(map:)1378 3563 y Fm(x)c Fl(7!)g Fm(xy)s(z)t(;)37 b(y)25 b Fl(7!)e Fm(xy)s(z)t(;)37 b(z)26 b Fl(7!)d Fm(x)2332 3529 y Fk(3)2370 3563 y Fm(y)2414 3529 y Fk(3)2451 3563 y Fm(z)t(:)639 3727 y Fn(\(iii\))31 b(Since)f Fm(z)1061 3739 y Fi(n)1106 3727 y Fn(\()p Fm(E)5 b Fn(\))30 b(corresp)r(onds)e (to)i(the)g(ev)n(olution)f(of)h(the)g(traces)f(asso)r(ciated)g(with)515 3827 y(the)39 b(w)n(ord)e Fm(\015)928 3839 y Fk(0)1007 3827 y Fn(=)k Fm(ab)d Fn(whic)n(h)h(con)n(tains)e Fm(a)p Fn(,)42 b(Lemma)c(3.4)g(of)h([9)o(])g(sho)n(ws)e(that)i(for)f(ev)n(ery) 515 3926 y Fm(E)g Fl(2)33 b Fn(\006,)i(there)e(is)h(a)f(subsequence)g (of)g(\()p Fl(j)p Fm(z)1872 3938 y Fi(n)1917 3926 y Fn(\()p Fm(E)5 b Fn(\))p Fl(j)p Fn(\))2102 3938 y Fi(n)p Fe(2)p Fb(N)2269 3926 y Fn(whic)n(h)33 b(is)g(b)r(ounded)h(from)f(ab)r(o)n(v)n (e.)515 4026 y(Then)38 b(the)h(ab)r(o)n(v)n(e)e(recursions)g(for)h Fm(x)1727 4038 y Fi(n)1813 4026 y Fn(=)j Fm(y)1960 4038 y Fi(n)2043 4026 y Fn(sho)n(w)d(that)h(the)f(sequence)g(\()p Fm(x)3038 4038 y Fi(n)3084 4026 y Fn(\()p Fm(E)5 b Fn(\)\))3246 4038 y Fi(n)p Fe(2)p Fb(N)515 4126 y Fn(cannot)27 b(con)n(v)n(erge)e (to)j(0.)639 4225 y(Th)n(us,)d(b)n(y)e(Corollary)f(2,)i(there)g(is)g Fm(!)i Fl(2)d Fn(\012)1939 4237 y Fi(u)2007 4225 y Fn(suc)n(h)g(that)i Fm(u)e Fn(is)h(the)g(restriction)f(of)h Fm(!)j Fn(to)d Fj(N)515 4325 y Fn(and)29 b(for)f(ev)n(ery)g(non-constan)n(t)g Fm(f)9 b Fn(,)29 b(the)h(sp)r(ectrum)f(of)g Fm(H)36 b Fn(with)30 b Fm(V)19 b Fn(\()p Fm(n)p Fn(\))26 b(=)f Fm(f)9 b Fn(\()p Fm(!)2959 4337 y Fi(n)3004 4325 y Fn(\))29 b(is)g(purely)515 4425 y(singular)d(con)n(tin)n(uous)h(and)g(supp)r (orted)h(on)f(a)g(Can)n(tor)f(set)i(of)f(Leb)r(esgue)h(measure)e(zero.) 515 4654 y Fc(3.3)112 b(A)37 b(family)f(of)i(examples)515 4807 y Fn(W)-7 b(e)28 b(can)g(v)n(erify)g(the)h(assumptions)e(of)i (Corollary)c(2)j(for)g(a)g(class)f(of)i(t)n(w)n(o-letter)e(substitu-) 515 4907 y(tions,)j(namely)g(all)f(those)h(whic)n(h)g(are)f(de\014ned)h (b)n(y)g Fm(S)5 b Fn(\()p Fm(a)p Fn(\))27 b(=)i Ff(p)l(alindr)l(ome)34 b(of)f(even)f(length)515 5006 y(b)l(e)l(ginning)e(with)e Fm(a)p Fn(,)g Fm(S)5 b Fn(\()p Fm(b)p Fn(\))23 b(=)f Fm(a)1463 4976 y Fi(p)1501 5006 y Fn(,)28 b Fm(p)23 b Fl(2)g Fj(N)t Fn(.)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 639 523 a Fn(First,)49 b(w)n(e)c(remark)e(that)i(this)g(class)e (clearly)h(v)n(eri\014es)f(the)j(h)n(yp)r(othesis)e(of)g(semi-)515 623 y(primitivit)n(y)-7 b(.)70 b(Second,)41 b(it)e(is)f(easy)g(to)g (see)h(in)f(this)h(case)f(that)h(if)g(for)f(some)g Fm(E)44 b Fn(in)39 b(the)515 722 y(sp)r(ectrum,)34 b(the)f(sequence)g(\()p Fm(x)1484 734 y Fi(n)1530 722 y Fn(\()p Fm(E)5 b Fn(\)\))1692 734 y Fi(n)p Fe(2)p Fb(N)1858 722 y Fn(con)n(v)n(erges)30 b(to)j(0,)h(then)f(for)f Fm(y)2808 734 y Fi(n)2853 722 y Fn(\()p Fm(E)5 b Fn(\))p Fm(;)14 b(z)3059 734 y Fi(n)3105 722 y Fn(\()p Fm(E)5 b Fn(\))33 b(de-)515 822 y(\014ned)28 b(as)e(in)i(the)g(previous)e(subsection,)i(the)g(sequence)e(\()p Fl(j)p Fm(y)2397 834 y Fi(n)2443 822 y Fn(\()p Fm(E)5 b Fn(\))p Fl(j)p Fn(\))2628 834 y Fi(n)p Fe(2)p Fb(N)2788 822 y Fn(con)n(v)n(erges)25 b(to)i(0)h(if)515 922 y Fm(p)h Fn(is)g(o)r(dd)g(and)g(2)f(if)i Fm(p)f Fn(is)g(ev)n(en.)41 b(Since)29 b(there)g(is)g(a)f(subsequence)h(of)g(\()p Fl(j)p Fm(z)2777 934 y Fi(n)2822 922 y Fn(\()p Fm(E)5 b Fn(\))p Fl(j)p Fn(\))3007 934 y Fi(n)p Fe(2)p Fb(N)3169 922 y Fn(whic)n(h)515 1021 y(is)31 b(b)r(ounded)h(from)g(ab)r(o)n(v)n (e,)f(this)h(leads)f(immediately)h(to)g(a)f(con)n(tradiction)f(b)r (ecause)i(the)515 1121 y(sp)r(eci\014c)d(form)g(of)g Fm(S)5 b Fn(\()p Fm(a)p Fn(\))30 b(implies)f(that)h(the)f(trace)g(map)g (expression)f(for)g Fm(x)2875 1133 y Fi(n)2921 1121 y Fn(\()p Fm(E)5 b Fn(\))30 b(con)n(tains)515 1220 y(a)d(constan)n(t)g (term)g(of)h(absolute)f(v)-5 b(alue)27 b(2)g(and)h(no)f(term)h Fm(y)2340 1232 y Fi(n)2385 1220 y Fn(\()p Fm(E)5 b Fn(\))28 b(or)f Fl(\000)p Fm(y)2751 1232 y Fi(n)2795 1220 y Fn(\()p Fm(E)5 b Fn(\).)639 1320 y(A)29 b(natural)f(goal)f(is)h(no)n(w)g(the)h (generalization)d(of)j(this)f(result)h(to)f(an)g(arbitrary)f(\014nite) 515 1420 y(alphab)r(et)c Fl(A)g Fn(=)g Fl(f)p Fm(a)1117 1432 y Fk(1)1153 1420 y Fm(;)14 b(:::::;)g(a)1386 1432 y Fi(m)1449 1420 y Fl(g)p Fn(,)24 b(in)f(the)g(sense)g(that)g Fm(S)5 b Fn(\()p Fm(a)2284 1432 y Fk(1)2321 1420 y Fn(\))23 b(=)g Ff(p)l(alindr)l(ome)k(of)f(even)g(length)515 1519 y(b)l(e)l(ginning)f(with)f Fm(a)1093 1531 y Fk(1)1153 1519 y Fn(and)f(all)g(the)h(other)e Fm(S)5 b Fn(\()p Fm(a)1904 1531 y Fi(k)1945 1519 y Fn(\)'s)23 b(are)f(p)r(o)n(w)n(ers)f (of)i Fm(a)2594 1531 y Fk(1)2632 1519 y Fn(,)h(b)r(ecause)e(in)i(this)f (case)515 1619 y(the)33 b(con)n(v)n(ergence)e(of)h(\()p Fm(x)1306 1631 y Fi(n)1352 1619 y Fn(\()p Fm(E)5 b Fn(\)\))1514 1631 y Fi(n)p Fe(2)p Fb(N)1680 1619 y Fn(to)33 b(0)f(implies)i(the)f (con)n(v)n(ergence)d(to)j(0,)h(2,)g(or)e Fl(\000)p Fn(2)g(of)515 1719 y(all)c(the)i(other)e(sequences)h(of)g(traces)f(asso)r(ciated)f (to)i(one)g(letter,)g(while)h Fm(x)2876 1731 y Fi(n)2921 1719 y Fn(\()p Fm(E)5 b Fn(\))30 b(con)n(tains)515 1818 y(a)f(constan)n(t)g(term)g(of)h(absolute)e(v)-5 b(alue)30 b(2)f(and)g(no)h(term)f(of)g(the)h(t)n(yp)r(e)g Fm(y)2789 1830 y Fi(n)2834 1818 y Fn(\()p Fm(E)5 b Fn(\))30 b(or)f Fl(\000)p Fm(y)3204 1830 y Fi(n)3248 1818 y Fn(\()p Fm(E)5 b Fn(\))515 1918 y(asso)r(ciated)26 b(to)h(other)g(letters.)515 2150 y Fc(3.4)112 b(Examples)37 b(with)f(an)i(in)m(v)-6 b(arian)m(t)515 2303 y Fn(W)f(e)32 b(ha)n(v)n(e)e(seen)h(that)h(for)f (an)g(attempt)i(to)e(apply)g(Theorem)g(3,)h(the)g(hardest)f(part)g(is)g (to)515 2403 y(establish)f(condition)g(\(iii\))h(since)f(condition)g (\(i\))h(can)f(b)r(e)h(v)n(eri\014ed)f(easily)f(and)h(condition)515 2503 y(\(ii\))21 b(can)f(b)r(e)h(c)n(hec)n(k)n(ed)e(b)n(y)i(a)f(simple) g(algorithmic)g(pro)r(cedure;)i(compare)d([9)o(].)35 b(W)-7 b(e)21 b(therefore)515 2602 y(w)n(an)n(t)d(to)g(p)r(oin)n(t)h (out)g(that)g(condition)f(\(iii\))i(can)e(sometimes)g(b)r(e)i (established)e(b)n(y)g(using)h(some)515 2702 y(soft)28 b(argumen)n(ts)g(in)n(v)n(olving)f(a)h(trace)g(map)h(in)n(v)-5 b(arian)n(t.)39 b(F)-7 b(or)28 b(example,)g(in)h(the)g(Fib)r(onacci)515 2813 y(case)d(w)n(e)i(ha)n(v)n(e)e(that)i Fm(x)1231 2825 y Fi(n)1277 2813 y Fn(\()p Fm(E)5 b Fn(\))23 b(=)g(tr\()p Fm(M)1705 2770 y Fk(\()p Fi(n)p Fk(\))1696 2838 y Fi(E)1802 2813 y Fn(\()p Fm(a)p Fn(\)\))28 b(ob)r(eys)696 2996 y(\()p Fm(x)775 3008 y Fi(n)p Fk(+1)905 2996 y Fn(\()p Fm(E)5 b Fn(\)\))1067 2961 y Fk(2)1124 2996 y Fn(+)18 b(\()p Fm(x)1286 3008 y Fi(n)1332 2996 y Fn(\()p Fm(E)5 b Fn(\)\))1494 2961 y Fk(2)1551 2996 y Fn(+)18 b(\()p Fm(x)1713 3008 y Fi(n)p Fe(\000)p Fk(1)1844 2996 y Fn(\()p Fm(E)5 b Fn(\)\))2006 2961 y Fk(2)2062 2996 y Fl(\000)18 b Fm(x)2192 3008 y Fi(n)p Fk(+1)2322 2996 y Fn(\()p Fm(E)5 b Fn(\))p Fm(x)2499 3008 y Fi(n)2545 2996 y Fn(\()p Fm(E)g Fn(\))p Fm(x)2722 3008 y Fi(n)p Fe(\000)p Fk(1)2854 2996 y Fn(\()p Fm(E)g Fn(\))23 b(=)g Fm(C)3154 3008 y Fi(f)515 3178 y Fn(for)31 b(ev)n(ery)g Fm(n)g Fn(and)h Fm(E)5 b Fn(,)33 b(with)g Fm(C)1492 3190 y Fi(f)1565 3178 y Fl(6)p Fn(=)d(0)h(if)i Fm(f)40 b Fn(is)32 b(not)g(constan)n(t)f([34)o (].)50 b(It)32 b(is)g(clear)f(that)h(this)515 3278 y(in)n(v)-5 b(arian)n(t)29 b(prev)n(en)n(ts)f Fm(x)1245 3290 y Fi(n)1291 3278 y Fn(\()p Fm(E)5 b Fn(\))31 b(from)e(con)n(v)n(erging)f(to)h (zero!)43 b(Mo)r(dels)30 b(with)g(in)n(v)-5 b(arian)n(t)29 b(ha)n(v)n(e)515 3378 y(b)r(een)e(discussed)f(in)g([3,)h(4)o(,)g(14)o (,)f(31)o(,)h(32)o(].)37 b(W)-7 b(e)26 b(remark)f(that)i(also)e(in)i (the)g(p)r(erio)r(d)f(doubling)515 3477 y(case,)h(an)g(in)n(v)-5 b(arian)n(t)26 b(pla)n(ys)h(an)g(imp)r(ortan)n(t)h(role)e(\(this)i(is)g (implicit)g(in)g([6]\).)515 3627 y Ff(A)l(cknow)t(le)l(dgements.)56 b Fn(D.)34 b(D.)h(w)n(ould)e(lik)n(e)g(to)h(thank)f(the)h(Cen)n(tre)f (de)h(Ph)n(ysique)f(Th)n(\023)-39 b(eo-)515 3726 y(rique,)24 b(Marseille)g(for)g(its)g(w)n(arm)g(hospitalit)n(y)g(and)g(the)h(Univ)n (ersit)n(\023)-39 b(e)23 b(de)h(T)-7 b(oulon)24 b(et)h(du)g(V)-7 b(ar)515 3826 y(and)33 b(the)i(German)e(Academic)h(Exc)n(hange)e (Service)h(\(HSP)h(I)r(I)r(I,)h(P)n(ostdoktoranden\))d(for)515 3926 y(\014nancial)27 b(supp)r(ort.)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fo(References)556 705 y Fn([1])41 b(J.-P)-7 b(.)37 b(Allouc)n(he,)k(Sc)n(hr\177)-42 b(odinger)36 b(op)r(erators)g(with)i(Rudin-Shapiro)f(p)r(oten)n(tials)h(are)685 805 y(not)28 b(palindromic,)f Ff(J.)j(Math.)h(Phys.)e Fg(38)e Fn(\(1997\),)f(1843{1848)556 971 y([2])41 b(M.)26 b(Baak)n(e,)e(A)i(note)f(on)g(palindromicit)n(y)-7 b(,)25 b Ff(L)l(ett.)i(Math.)i(Phys.)e Fg(49)e Fn(\(1999\),)f(217{227)556 1137 y([3])41 b(M.)27 b(Baak)n(e,)f(U.)h(Grimm,)g(and)g(D.)g(Joseph,)f (T)-7 b(race)26 b(maps,)h(in)n(v)-5 b(arian)n(ts,)25 b(and)i(some)f(of)685 1236 y(their)i(applications,)f Ff(Int.)i(J.)h(Mo)l(d.)h(Phys.)g(B)d Fg(7)g Fn(\(1993\),)e(1527{1550) 556 1402 y([4])41 b(M.)35 b(Baak)n(e)e(and)i(J.)g(A.)g(G.)g(Rob)r (erts,)h(Rev)n(ersing)d(symmetry)i(group)e(of)i Fm(GL)p Fn(\(2)p Fm(;)14 b Fj(Z)o Fn(\))685 1502 y(and)29 b Fm(P)12 b(GL)p Fn(\(2)p Fm(;)i Fj(Z)o Fn(\))23 b(matrices)28 b(with)h(connections)f(to)h(cat)g(maps)f(and)h(trace)f(maps,)h Ff(J.)685 1602 y(Phys.)j(A)27 b Fg(30)g Fn(\(1997\),)f(1549{1573)556 1768 y([5])41 b(J.)35 b(Bellissard,)h(Sp)r(ectral)f(prop)r(erties)f(of) h(Sc)n(hr\177)-42 b(odinger's)34 b(op)r(erator)f(with)j(a)f(Th)n(ue-) 685 1867 y(Morse)40 b(p)r(oten)n(tial,)k(in:)64 b Ff(Numb)l(er)41 b(The)l(ory)i(and)g(Physics)f Fn(\(Les)f(Houc)n(hes,)j(1989\),)685 1967 y(Eds.)27 b(J.)f(M.)h(Luc)n(k,)g(P)-7 b(.)26 b(Moussa)g(and)h(M.)g (W)-7 b(aldsc)n(hmidt,)27 b(Springer,)f(Berlin)g(\(1990\),)685 2066 y(pp.)i(140{150)556 2232 y([6])41 b(J.)20 b(Bellissard,)g(A.)h(Bo) n(vier,)f(and)g(J.-M.)f(Ghez,)j(Sp)r(ectral)e(prop)r(erties)f(of)h(a)g (tigh)n(t)g(bind-)685 2332 y(ing)39 b(Hamiltonian)g(with)g(p)r(erio)r (d)g(doubling)f(p)r(oten)n(tial,)k Ff(Commun.)e(Math.)i(Phys.)685 2432 y Fg(135)27 b Fn(\(1991\),)g(379{399)556 2598 y([7])41 b(J.)31 b(Bellissard,)e(A.)i(Bo)n(vier,)f(and)g(J.-M.)g(Ghez,)i(Gap)e (lab)r(elling)g(theorems)g(for)g(one-)685 2697 y(dimensional)40 b(discrete)g(Sc)n(hr\177)-42 b(odinger)38 b(op)r(erators,)j Ff(R)l(ev.)g(Math.)i(Phys.)e Fg(4)f Fn(\(1992\),)685 2797 y(1{37)556 2963 y([8])h(J.)28 b(Bellissard,)e(B.)h(Io)r(c)n(h)n (um,)g(E.)g(Scopp)r(ola,)g(and)g(D.)h(T)-7 b(estard,)27 b(Sp)r(ectral)g(prop)r(erties)685 3063 y(of)e(one-dimensional)e (quasi-crystals,)g Ff(Commun.)28 b(Math.)g(Phys.)e Fg(125)e Fn(\(1989\),)g(527{)685 3162 y(543)556 3328 y([9])41 b(A.)34 b(Bo)n(vier)e(and)h(J.-M.)g(Ghez,)i(Sp)r(ectral)e(prop)r (erties)f(of)h(one-dimensional)f(Sc)n(hr\177)-42 b(o-)685 3428 y(dinger)46 b(op)r(erators)e(with)i(p)r(oten)n(tials)g(generated)f (b)n(y)h(substitutions,)51 b Ff(Commun.)685 3528 y(Math.)e(Phys.)f Fg(158)e Fn(\(1993\),)k(45{66;)k(Erratum)45 b Ff(Commun.)j(Math.)h (Phys.)f Fg(166)685 3627 y Fn(\(1994\),)27 b(431{432)515 3793 y([10])40 b(D.)52 b(Damanik,)57 b Fm(\013)p Fn(-con)n(tin)n(uit)n (y)50 b(prop)r(erties)g(of)h(one-dimensional)f(quasicrystals,)685 3893 y Ff(Commun.)30 b(Math.)h(Phys.)f Fg(192)c Fn(\(1998\),)h(169{182) 515 4059 y([11])40 b(D.)26 b(Damanik,)f(Singular)f(con)n(tin)n(uous)g (sp)r(ectrum)i(for)e(the)i(p)r(erio)r(d)e(doubling)h(Hamil-)685 4159 y(tonian)d(on)g(a)g(set)h(of)f(full)h(measure,)f Ff(Commun.)k(Math.)g(Phys.)e Fg(196)d Fn(\(1998\),)h(477{483)515 4325 y([12])40 b(D.)47 b(Damanik,)k(Singular)46 b(con)n(tin)n(uous)f (sp)r(ectrum)i(for)f(a)g(class)f(of)i(substitution)685 4424 y(Hamiltonians,)28 b Ff(L)l(ett.)h(Math.)i(Phys.)e Fg(46)e Fn(\(1998\),)g(303{311)515 4590 y([13])40 b(D.)55 b(Damanik,)61 b(Gordon-t)n(yp)r(e)54 b(argumen)n(ts)f(in)i(the)g(sp)r (ectral)f(theory)f(of)i(one-)685 4690 y(dimensional)34 b(quasicrystals,)h(to)f(app)r(ear)f(in)i Ff(Dir)l(e)l(ctions)h(in)h (Mathematic)l(al)h(Qua-)685 4790 y(sicrystals)p Fn(,)24 b(Eds.)d(M.)g(Baak)n(e)f(and)h(R.)h(V.)f(Mo)r(o)r(dy)-7 b(,)23 b(CRM)e(Monograph)f(Series,)h(AMS,)685 4889 y(Pro)n(vidence)26 b(\(2000\))1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fn([14])40 b(D.)24 b(Damanik,)g(Substitution)g (Hamiltonians)e(with)i(b)r(ounded)g(trace)e(map)h(orbits,)g(to)685 623 y(app)r(ear)k(in)h Ff(J.)i(Math.)h(A)n(nal.)f(Appl.)515 789 y Fn([15])40 b(D.)21 b(Damanik,)h(Uniform)f(singular)e(con)n(tin)n (uous)g(sp)r(ectrum)i(for)f(the)h(p)r(erio)r(d)f(doubling)685 888 y(Hamiltonian,)28 b(preprin)n(t)f(\(2000\))515 1054 y([16])40 b(D.)47 b(Damanik,)k(Singular)46 b(con)n(tin)n(uous)f(sp)r (ectrum)i(for)f(a)g(class)f(of)i(substitution)685 1154 y(Hamiltonians)28 b(I)r(I.,)g(preprin)n(t)f(\(2000\))515 1320 y([17])40 b(D.)34 b(Damanik,)f(R.)g(Killip,)h(and)f(D.)g(Lenz,)h (Uniform)e(sp)r(ectral)g(prop)r(erties)g(of)h(one-)685 1420 y(dimensional)40 b(quasicrystals,)i(I)r(I)r(I.)e Fm(\013)p Fn(-con)n(tin)n(uit)n(y)-7 b(,)44 b Ff(Commun.)d(Math.)i (Phys.)f Fg(212)685 1519 y Fn(\(2000\),)27 b(191{204)515 1685 y([18])40 b(D.)31 b(Damanik)e(and)g(D.)h(Lenz,)g(Uniform)g(sp)r (ectral)f(prop)r(erties)g(of)g(one-dimensional)685 1785 y(quasicrystals,)21 b(I.)h(Absence)g(of)f(eigen)n(v)-5 b(alues,)22 b Ff(Commun.)j(Math.)g(Phys.)e Fg(207)e Fn(\(1999\),)685 1885 y(687{696)515 2051 y([19])40 b(D.)i(Damanik)f(and)h(D.)g(Zare,)h (P)n(alindrome)d(complexit)n(y)h(b)r(ounds)g(for)g(primitiv)n(e)685 2150 y(substitution)29 b(sequences,)d Ff(Discr)l(ete)k(Math.)f Fg(222)e Fn(\(2000\),)f(259{267)515 2316 y([20])40 b(F.)26 b(Dely)n(on)f(and)g(D.)h(P)n(etritis,)f(Absence)g(of)g(lo)r(calization) f(in)i(a)e(class)h(of)g(Sc)n(hr\177)-42 b(odinger)685 2416 y(op)r(erators)23 b(with)j(quasip)r(erio)r(dic)e(p)r(oten)n(tial,) i Ff(Commun.)i(Math.)g(Phys.)f Fg(103)d Fn(\(1986\),)685 2516 y(441{444)515 2682 y([21])40 b(F.)30 b(Dely)n(on,)g(J.)f(P)n(eyri) n(\022)-39 b(ere,)27 b(Recurrence)i(of)g(the)h(eigenstates)e(of)i(a)f (Sc)n(hr\177)-42 b(odinger)28 b(op-)685 2781 y(erator)e(with)i (automatic)f(p)r(oten)n(tial,)h Ff(J.)i(Stat.)f(Phys.)g Fg(64)f Fn(\(1991\),)e(363{368)515 2947 y([22])40 b(A.)i(Gordon,)h(On)d (the)i(p)r(oin)n(t)f(sp)r(ectrum)f(of)h(the)g(one-dimensional)f(Sc)n (hr\177)-42 b(odinger)685 3047 y(op)r(erator,)26 b Ff(Usp.)31 b(Math.)g(Nauk)c Fg(31)g Fn(\(1976\),)g(257{258)515 3213 y([23])40 b(M.)22 b(H\177)-42 b(ornquist)22 b(and)f(M.)i(Johansson,)e (Singular)g(con)n(tin)n(uous)g(electron)g(sp)r(ectrum)h(for)685 3313 y(a)28 b(class)e(of)i(circle)f(sequences,)g Ff(J.)i(Phys.)j(A)27 b Fg(28)g Fn(\(1995\),)f(479{495)515 3479 y([24])40 b(A.)34 b(Hof,)h(O.)e(Knill,)i(and)e(B.)g(Simon,)i(Singular)d(con)n(tin)n(uous) g(sp)r(ectrum)i(for)e(palin-)685 3578 y(dromic)24 b(Sc)n(hr\177)-42 b(odinger)23 b(op)r(erators,)g Ff(Commun.)k(Math.)h(Phys.)e Fg(174)e Fn(\(1995\),)f(149{159)515 3744 y([25])40 b(S.)c(Jitomirsk)-5 b(a)n(y)n(a)33 b(and)i(B.)g(Simon,)i(Op)r(erators)c(with)j(singular)e (con)n(tin)n(uous)g(sp)r(ec-)685 3844 y(trum:)44 b(I)r(I)r(I.)32 b(Almost)f(p)r(erio)r(dic)g(Sc)n(hr\177)-42 b(odinger)29 b(op)r(erators,)h Ff(Commun.)j(Math.)i(Phys.)685 3944 y Fg(165)27 b Fn(\(1994\),)g(201{205)515 4110 y([26])40 b(S.)e(Kotani,)i(Jacobi)d(matrices)g(with)h(random)f(p)r(oten)n(tials)g (taking)g(\014nitely)i(man)n(y)685 4209 y(v)-5 b(alues,)28 b Ff(R)l(ev.)i(Math.)h(Phys.)e Fg(1)e Fn(\(1989\),)f(129{133)515 4375 y([27])40 b(Y.)c(Last)e(and)g(B.)h(Simon,)i(Eigenfunctions,)f (transfer)e(matrices,)h(and)g(absolutely)685 4475 y(con)n(tin)n(uous)45 b(sp)r(ectrum)g(of)h(one-dimensional)e(Sc)n(hr\177)-42 b(odinger)43 b(op)r(erators,)48 b Ff(Invent.)685 4575 y(Math.)29 b Fg(135)e Fn(\(1999\),)g(329{367)515 4741 y([28])40 b(J.-M.)25 b(Luc)n(k,)h(Can)n(tor)d(sp)r(ectra)i(and)g (scaling)f(of)h(gap)g(widths)h(in)f(deterministic)g(ap)r(e-)685 4840 y(rio)r(dic)j(systems,)f Ff(Phys.)k(R)l(ev.)f(B)e Fg(39)f Fn(\(1989\),)f(5834{5849)1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fn([29])40 b(M.)28 b(Que\013)n(\023)-39 b(elec,)26 b Ff(Substitution)j(Dynamic)l(al)h(Systems)f(-)h(Sp)l(e)l (ctr)l(al)f(A)n(nalysis)p Fn(,)f(Lecture)685 623 y(Notes)j(in)g (Mathematics,)g(V)-7 b(ol.)31 b(1284,)e(Springer,)i(Berlin,)g(Heidelb)r (erg,)g(New)g(Y)-7 b(ork)685 722 y(\(1987\))515 888 y([30])40 b(L.)28 b(Ra)n(ymond,)f(A)h(constructiv)n(e)e(gap)h(lab)r(elling)h(for) f(the)h(discrete)f(Sc)n(hr\177)-42 b(odinger)26 b(op-)685 988 y(erator)g(on)i(a)f(quasip)r(erio)r(dic)g(c)n(hain,)g(preprin)n(t)g (\(1997\))515 1154 y([31])40 b(J.)35 b(A.)f(G.)h(Rob)r(erts,)h (Escaping)c(orbits)i(in)h(trace)e(maps,)j Ff(Physic)l(a)i(A)c Fg(228)f Fn(\(1996\),)685 1254 y(295{325)515 1420 y([32])40 b(J.)34 b(A.)g(G.)h(Rob)r(erts)e(and)h(M.)g(Baak)n(e,)g(T)-7 b(race)33 b(maps)g(as)h(3D)f(rev)n(ersible)g(dynamical)685 1519 y(systems)27 b(with)i(an)e(in)n(v)-5 b(arian)n(t,)26 b Ff(J.)k(Stat.)g(Phys.)f Fg(74)e Fn(\(1994\),)f(829{888)515 1685 y([33])40 b(D.)32 b(Shec)n(h)n(tman,)f(I.)g(Blec)n(h,)g(D.)h (Gratias,)e(and)h(J.)g(V.)g(Cahn,)g(Metallic)g(phase)g(with)685 1785 y(long-range)e(orien)n(tational)h(order)g(and)h(no)g (translational)f(symmetry)-7 b(,)32 b Ff(Phys.)j(R)l(ev.)685 1885 y(L)l(ett.)28 b Fg(53)f Fn(\(1984\),)f(1951{1953)515 2051 y([34])40 b(A.)28 b(S)r(\177)-44 b(ut})i(o,)27 b(The)g(sp)r (ectrum)g(of)g(a)f(quasip)r(erio)r(dic)g(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erator,)h Ff(Commun.)685 2150 y(Math.)31 b(Phys.)f Fg(111)c Fn(\(1987\),)h(409{415)515 2316 y([35])40 b(A.)34 b(S)r(\177)-44 b(ut})i(o,)35 b(Singular)d(con)n(tin)n(uous)g (sp)r(ectrum)i(on)e(a)h(Can)n(tor)f(set)h(of)h(zero)e(Leb)r(esgue)685 2416 y(measure)27 b(for)g(the)h(Fib)r(onacci)f(Hamiltonian,)h Ff(J.)i(Stat.)f(Phys.)g Fg(56)e Fn(\(1989\),)g(525{531)1905 5255 y(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF