The following PostScript-file was created out of a Latex-source. But, unfortunately, it was no good distributing this Latex-text because there are 22 figures which have to be bound in (as PostScript files), which occupy the main part of storage; so there would be no advantage in splitting everything up into different pieces... Nevertheless we tried to keep the size of this file as small as possible by reducing the quality of the pictures. This way, all included figures have mere "Fax-quality" (72dpi) so they really look a little bit poor, but we think it will be still possible to grasp everything that we want to say; if there should be problems with this file (or the contents(?!)), or if there should be the need of having text or pictures in other formats, please don't hesitate to contact us: E-mail: gawl@atlas.tp1.ruhr-uni-bochum.de E-mail: exner@vax.ujf.cas.cz BODY %!PS-Adobe-2.0 %%Creator: dvips 5.519 Copyright 1986, 1993 Radical Eye Software %%Title: rectlat.dvi %%CreationDate: Mon Aug 21 13:29:32 1995 %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -Php4 rectlat.dvi %DVIPSSource: TeX output 1995.08.16:2035 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 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b(coupling)g(constan)m(ts)h (could)f(corresp)s(ond)g(to)g(lo)s(cal)f(deformation)g(of)g(the)h (junction)165 4808 y(region,)30 b(impurities)g(or)e(in\015uence)i(of)e (external)h(\014elds)h(|)e(in)g(short,)i Fo(imp)-5 b(erfe)g(ct)30 b(c)-5 b(ontacts)28 b Fw(|)g(no)165 4928 y(con)m(vicing)34 b(answ)m(er)g(is)g(kno)m(wn)f(up)g(to)g(no)m(w.)331 5049 y(Moreo)m(v)m(er,)h(the)f(coupling)g(\(Re)16 b(fdelta)34 b(b)s(c\))e(is)i(not)e(the)h(only)g(p)s(ossible:)45 b(for)33 b(a)f(junction)h(of)165 5169 y Fn(N)60 b Fw(links)33 b(there)g(is)f(in)h(general)f(an)49 b Fn(N)1568 5133 y Fr(2)1640 5169 y Fw(parameter)32 b(family)h(of)f(self{adjoin)m(t)g (op)s(erators)g(whic)m(h)165 5289 y(act)39 b(as)f(the)h(free)g (Hamiltonian)f(outside)i(the)f(branc)m(hing)g(p)s(oin)m(ts.)61 b(A)39 b(metho)s(d)f(to)h(construct)165 5410 y(suc)m(h)26 b(op)s(erators)e(and)h(some)g(particular)f(classes)j(of)c(them)i(w)m (ere)h(discussed)h(in)e(detail)f(in)h(Ref.14.)1913 5682 y(3)p eop %%Page: 4 4 4 3 bop 165 218 a Fw(In)35 b(distinction)h(to)e(\(Re)16 b(fdelta)34 b(b)s(c\),)h(the)g(\\additional")f(couplings)h(ha)m(v)m(e)h (w)m(a)m(v)m(efunctions)h Fo(dis-)165 338 y(c)-5 b(ontinuous)30 b Fw(at)g(the)g(v)m(ertex,)j Fo(i.e.)p Fw(,)d(the)g(limits)h(for)f(at)f (least)i(one)f(pair)g(of)g(links)h(di\013er)f(m)m(utually)165 459 y(there.)331 579 y(This)45 b(feature)f(is)g(not)g(automatically)g (disqualifying.)79 b(It)44 b(w)m(as)h(sho)m(wn)g(in)f(Ref.10)g(that)165 699 y(the)31 b(so{called)47 b Fn(\016)804 663 y Fq(0)857 699 y Fw(in)m(teraction)1314 663 y Fr(17)1420 699 y Fw(represen)m(ts)32 b(a)e(reasonable)g(\(if)g(idealized\))h(mo)s(del)g(for)e(a)h(com-)165 820 y(plicated)39 b(geometric)f(scatterer,)j(in)c(whic)m(h)j(instead)e (b)m(y)h(a)e(p)s(oin)m(t)h(con)m(tact)g(t)m(w)m(o)h(hal\015ines)g(are) 165 940 y(joined)d(b)m(y)h(n)m(umerous)g(short)e(\\wires".)54 b(Moreo)m(v)m(er,)37 b(this)g(result)f(extends)h(to)f(junctions)g(with) 165 1060 y(an)m(y)42 b(n)m(um)m(b)s(er)h(of)d(links;)1065 1024 y Fr(13)1183 1060 y Fw(the)h(most)h(natural)f(coun)m(terpart)h(to) f(\(Re)16 b(fdelta)41 b(b)s(c\))g(app)s(ears)h(to)165 1181 y(b)s(e)f(the)g(so{called)58 b Fn(\016)966 1145 y Fq(0)962 1205 y Fv(s)1056 1181 y Fo(inter)-5 b(action)40 b Fw(whic)m(h)i(requires)g(the)f(w)m(a)m(v)m(efunction)i Fo(derivatives)c Fw(to)h(b)s(e)165 1301 y(con)m(tin)m(uous,)51 b Fn( )764 1265 y Fq(0)760 1326 y Fr(1)800 1301 y Fw(\()p Fn(x)893 1316 y Fv(m)960 1301 y Fw(\))28 b(=)f Fm(\001)17 b(\001)g(\001)26 b Fw(=)i Fn( )1444 1265 y Fq(0)1440 1326 y Fv(N)1507 1301 y Fw(\()p Fn(x)1600 1316 y Fv(m)1667 1301 y Fw(\))g(=:)g Fn( )1931 1265 y Fq(0)1954 1301 y Fw(\()p Fn(x)2047 1316 y Fv(m)2114 1301 y Fw(\))17 b(,)32 b(and)1441 1438 y Fl(X)1485 1620 y Fv(j)1578 1521 y Fn( )1641 1536 y Fv(j)1677 1521 y Fw(\()p Fn(x)1770 1536 y Fv(m)1837 1521 y Fw(\))45 b(=)f Fn(\014)2095 1536 y Fv(m)2161 1521 y Fn( )2228 1480 y Fq(0)2252 1521 y Fw(\()p Fn(x)2345 1536 y Fv(m)2412 1521 y Fw(\))1134 b(\(2\))165 1819 y(for)41 b(some)59 b Fn(\014)648 1834 y Fv(m)732 1819 y Fw(.)70 b(The\\coupling)43 b(constan)m(ts")59 b Fn(\014)2006 1834 y Fv(m)2131 1819 y Fw(here)42 b(measure,)j(roughly)d(sp)s(eaking,) j(the)165 1939 y Fo(total)30 b(length)25 b Fw(of)h(the)h(wires)g(whic)m (h)h(constitute)f(the)g(geometric)g(scatterer;)j(for)25 b(a)h(more)h(detailed)165 2060 y(discussion)35 b(w)m(e)f(refer)f(to)f (Ref.13.)331 2180 y(In)43 b(what)h(follo)m(ws)g(w)m(e)h(shall)f(b)s(e)f (concerned)i(with)f(rectangular)g(lattices)h(in)e(whic)m(h)i(the)165 2300 y(coupling)30 b(at)f(eac)m(h)i(junction)f(is)g(the)f(same)i(and)e (b)s(elongs)h(to)f(one)h(the)g(ab)s(o)m(v)m(e)g(describ)s(ed)h(t)m(yp)s (es;)165 2421 y(for)25 b(the)g(sak)m(e)h(of)e(brevit)m(y)i(w)m(e)g (shall)f(refer)g(to)g(them)g(as)g(to)g(the)41 b Fn(\016)46 b Fw(and)41 b Fn(\016)2759 2385 y Fq(0)2755 2445 y Fv(s)2833 2421 y Fw(lattices,)28 b(resp)s(ectiv)m(ely)-8 b(.)165 2541 y(In)40 b(a)f(sense,)k(suc)m(h)d(lattices)h(represen)m(t)g(a)e (generalization)h(of)f(the)g(classical)i(Kronig{P)m(enney)165 2661 y(mo)s(del)33 b(and)g(its)50 b Fn(\016)845 2625 y Fq(0)900 2661 y Fw(mo)s(di\014cation)1428 2625 y Fr(2)1501 2661 y Fw(to)32 b(higher)h(dimensions.)2410 2625 y Fr(18)165 2994 y Fp(I)5 b(I)g(I)163 b(General)54 b(prop)5 b(erties)53 b(of)j(the)f(sp)5 b(ectra)165 3213 y Fw(Before)31 b(pro)s(ceeding)h (further)f(let)g(us)g(recall)g(some)h(results)g(ab)s(out)e(the)h(sp)s (ectra)h(of)e(the)h(consid-)165 3334 y(ered)j(lattice)f(Hamiltonians)g (deriv)m(ed)i(in)d(Ref.13.)165 3622 y Fx(I)t(I)t(I.1)158 b Fk(\016)70 b Fx(lattices)165 3807 y Fw(By)34 b(assumption,)h(a)50 b Fn(\016)j Fw(lattice)34 b(is)g(a)f(p)s(erio)s(dic)g(system)j(in)d(b)s (oth)g(directions.)47 b(P)m(erforming)34 b(the)165 3928 y(Blo)s(c)m(h)f(analysis,)h(w)m(e)g(arriv)m(e)f(at)g(the)g(band)f (condition)2180 3891 y Fr(19)855 4124 y Fw(cos)17 b Fn(#)1059 4139 y Fr(1)1099 4124 y Fn(`)1140 4139 y Fr(1)1202 4124 y Fm(\000)22 b Fw(cos)c Fn(k)s(`)1544 4139 y Fr(1)p 855 4168 728 4 v 1084 4260 a Fw(sin)f Fn(k)s(`)1315 4275 y Fr(1)1632 4191 y Fw(+)1756 4124 y(cos)h Fn(#)1961 4139 y Fr(2)2000 4124 y Fn(`)2041 4139 y Fr(2)2103 4124 y Fm(\000)23 b Fw(cos)17 b Fn(k)s(`)2445 4139 y Fr(2)p 1756 4168 V 1985 4260 a Fw(sin)g Fn(k)s(`)2216 4275 y Fr(2)2533 4191 y Fm(\000)2679 4124 y Fn(\013)p 2659 4168 103 4 v 2659 4260 a Fw(2)p Fn(k)2816 4191 y Fw(=)44 b(0)17 b Fn(:)555 b Fw(\(3\))165 4452 y(Although)34 b(an)f(analytic)h (solution)g(can)f(b)s(e)h(written)g(in)f(the)h(trivial)f(case)i(only)-8 b(,)33 b(the)h(condition)165 4572 y(\(Re)16 b(fdelta)34 b(band)f(condition\))h(nev)m(ertheless)j(allo)m(ws)d(to)f(dra)m(w)g (man)m(y)h(conclusions)h(ab)s(out)e(the)165 4693 y(sp)s(ectrum.)45 b(Let)33 b(us)g(rewrite)h(it)e(in)h(the)g(form)1466 4912 y Fn(\013)p 1446 4956 V 1446 5047 a Fw(2)p Fn(k)1603 4979 y Fw(=)1767 4871 y Fr(2)1725 4896 y Fl(X)1723 5078 y Fv(j)t Fr(=1)1889 4912 y Fn(v)1936 4927 y Fv(j)1995 4912 y Fm(\000)22 b Fw(cos)c Fn(k)s(`)2337 4927 y Fv(j)p 1889 4956 485 4 v 1997 5047 a Fw(sin)g Fn(k)s(`)2229 5062 y Fv(j)2411 4979 y Fw(;)165 5277 y(if)28 b(the)h(quasimomen)m(tum) h(comp)s(onen)m(ts)46 b Fn(#)1747 5292 y Fv(j)1801 5277 y Fn(;)e(j)34 b Fw(=)27 b(1)p Fn(;)17 b Fw(2)g(,)28 b(run)h(though)f (the)g(Brillouin)h(zone,)g(the)165 5397 y(ranges)34 b(of)f(the)h (parameters)51 b Fn(v)1320 5412 y Fv(j)1386 5397 y Fw(:=)30 b(cos)17 b Fn(#)1723 5412 y Fv(j)1760 5397 y Fn(`)1801 5412 y Fv(j)1887 5397 y Fw(co)m(v)m(er)35 b(the)f(in)m(terv)-5 b(al)51 b([)p Fm(\000)p Fw(1)p Fn(;)17 b Fw(1])g(.)46 b(It)34 b(is)g(easy)g(to)f(see)1913 5682 y(4)p eop %%Page: 5 5 5 4 bop 165 218 a Fw(that)33 b(for)f(a)g(\014xed)50 b Fn(k)20 b Fw(,)33 b(the)g(maxim)m(um)h(of)e(the)h(righ)m(t)g(side)g (equals)1138 504 y Fn(F)1201 519 y Fr(+)1260 504 y Fw(\()p Fn(k)s Fw(\))44 b(:=)1625 396 y Fr(2)1583 421 y Fl(X)1581 603 y Fv(j)t Fr(=1)1737 504 y Fw(tan)1894 358 y Fl( )1970 437 y Fn(k)s(`)2065 452 y Fv(j)p 1970 481 132 4 v 2012 572 a Fw(2)2150 504 y Fm(\000)2277 437 y Fn(\031)p 2277 481 59 4 v 2282 572 a Fw(2)2362 358 y Fl(")2420 437 y Fn(k)s(`)2515 452 y Fv(j)p 2420 481 132 4 v 2457 572 a Fn(\031)2562 358 y Fl(#!)2709 504 y Fn(;)165 802 y Fw(where)32 b(the)e(the)h(square)g(brac)m(k)m(et)h(denotes)f(con)m(v)m (en)m(tionally)i(the)d(in)m(teger)h(part,)g(and)f(the)g(mini-)165 922 y(m)m(um,)50 b Fn(F)518 937 y Fq(\000)578 922 y Fw(\()p Fn(k)s Fw(\))17 b(,)32 b(is)g(giv)m(en)i(b)m(y)f(a)f(similar)h(form)m (ula)f(with)50 b(tan)f(replaced)33 b(b)m(y)50 b Fm(\000)17 b Fw(cot)g(.)43 b(It)32 b(is)h(clear)165 1042 y(from)k(here)h(that)f (the)h(gaps)g(of)e(the)55 b Fn(\016)t Fw({lattice)37 b(sp)s(ectrum)i(with)f(a)f(coupling)g(constan)m(t)55 b Fn(\013)g Fw(on)165 1163 y(the)33 b(p)s(ositiv)m(e)h(part)f(of)f(the) h(energy)g(axis)h(are)e(determined)j(b)m(y)e(the)g(condition)1582 1400 y Fm(\006)1728 1333 y Fn(\013)p 1708 1377 103 4 v 1708 1469 a Fw(2)p Fn(k)1865 1400 y(>)44 b Fm(\006)p Fn(F)2125 1415 y Fq(\006)2185 1400 y Fw(\()p Fn(k)s Fw(\))1269 b(\(4\))165 1646 y(for)59 b Fm(\006)p Fn(\013)45 b(>)e Fw(0)17 b(,)44 b(resp)s(ectiv)m(ely)-8 b(.)75 b(The)43 b(negativ)m(e)h(part)e(of)f(the)i(sp)s(ectrum)g(is)g(obtained)g(analo-) 165 1766 y(gously)30 b(b)m(y)f(comparing)1109 1727 y Fv(\013)p 1093 1743 76 4 v 1093 1801 a Fr(2)p Fv(\024)1224 1766 y Fw(with)g(the)g(extrem)m(um)i(v)-5 b(alues)29 b(of)f(the)h(function)46 b Fn(iF)3111 1781 y Fq(\006)3170 1766 y Fw(\()p Fn(i\024)p Fw(\))17 b(.)42 b(Simple)165 1887 y(consequences)36 b(of)d(the)g(condition)g(\(Re)16 b(fdelta)33 b(gap\))f(are)g(the)h(follo)m(wing:)214 2090 y(\(a1\))44 b(The)h(sp)s(ectrum)h(has)f(a)f(band)h(structure;)51 b(it)45 b(coincides)h(with)f(the)g(p)s(ositiv)m(e)h(hal\015ine)409 2210 y([0)p Fn(;)17 b Fm(1)p Fw(\))48 b(if)33 b(and)g(only)g(if)49 b Fn(\013)28 b Fw(=)g(0)17 b(.)214 2337 y(\(a2\))35 b(If)53 b Fn(\013)34 b(>)f Fw(0)17 b(,)36 b(eac)m(h)g(upp)s(er)h(band)f(end)g (is)h(a)e(square)i(of)f(some)53 b Fn(k)2740 2352 y Fv(n)2820 2337 y Fw(:=)2966 2297 y Fv(\031)r(n)p 2966 2313 86 4 v 2977 2371 a(`)3006 2380 y Fj(1)3114 2337 y Fw(or)3256 2310 y(~)3253 2337 y Fn(k)3304 2352 y Fv(m)3404 2337 y Fw(:=)3550 2297 y Fv(\031)r(m)p 3550 2313 106 4 v 3571 2371 a(`)3600 2380 y Fj(2)3682 2337 y Fw(,)409 2457 y(where)g Fn(n;)33 b(m)52 b Fw(are)34 b(in)m(tegers.)52 b(Similarly)-8 b(,)36 b(for)50 b Fn(\013)32 b(<)f Fw(0)51 b(eac)m(h)36 b(lo)m(w)m(er)g(band)f(end,)h(starting)409 2577 y(from)d(the)g(second)g (one,)g(equals)51 b Fn(k)1695 2541 y Fr(2)1692 2602 y Fv(n)1788 2577 y Fw(or)1926 2551 y(~)1924 2577 y Fn(k)1978 2541 y Fr(2)1975 2602 y Fv(m)2057 2577 y Fw(.)214 2704 y(\(a3\))37 b(The)i(lo)m(w)m(est)g(band)f(threshold)h(is)g(p)s(ositiv)m (e)g(for)53 b Fn(\013)38 b(>)e Fw(0)54 b(and)38 b(negativ)m(e)h(if)54 b Fn(\013)37 b(>)g Fw(0)f(;)k(in)409 2824 y(the)35 b(case)52 b Fn(\013)32 b(<)f Fm(\000)p Fw(4\()p Fn(`)1210 2783 y Fq(\000)p Fr(1)1210 2846 y(1)1328 2824 y Fw(+)24 b Fn(`)1469 2783 y Fq(\000)p Fr(1)1469 2846 y(2)1563 2824 y Fw(\))51 b(the)35 b(whole)h(\014rst)f(band)g(is)g(negativ)m(e,)h(and) f(the)g(second)409 2973 y(one)e(starts)g(at)996 2877 y Fl(\020)1058 2934 y Fv(\031)p 1056 2950 48 4 v 1056 3008 a(L)1113 2877 y Fl(\021)1163 2900 y Fr(2)1219 2973 y Fw(,)g(where)50 b Fn(L)28 b Fw(:=)g(max\()p Fn(`)2062 2988 y Fr(1)2102 2973 y Fn(;)17 b(`)2187 2988 y Fr(2)2226 2973 y Fw(\))g(.)214 3112 y(\(a4\))32 b(The)i(p)s(ositiv)m(e)g(bands)f (shrink)h(with)f(increasing)50 b Fm(j)p Fn(\013)q Fm(j)17 b Fw(.)214 3238 y(\(a5\))23 b(Eac)m(h)h(gap)f(is)g(con)m(tained)i(in)e (the)h(in)m(tersection)h(of)e(a)f(pair)i(of)e(gaps)i(of)e(the)i (Kronig{P)m(enney)409 3359 y(mo)s(del)33 b(with)g(the)g(coupling)h (constan)m(t)50 b Fn(\013)f Fw(and)33 b(spacings)51 b Fn(`)2635 3374 y Fr(1)2723 3359 y Fw(and)e Fn(`)2970 3374 y Fr(2)3026 3359 y Fw(,)33 b(resp)s(ectiv)m(ely)-8 b(.)214 3485 y(\(a6\))35 b(All)g(gaps)h(ab)s(o)m(v)m(e)g(the)g (threshold)g(are)g(\014nite.)52 b(If)35 b(there)h(are)g(in\014nitely)h (man)m(y)f(of)f(them,)409 3605 y(their)c(widths)g(are)f(asymptotically) h(b)s(ounded)g(b)m(y)47 b(2)p Fm(j)p Fn(\013)q Fm(j)p Fw(\()p Fn(`)2557 3620 y Fr(1)2595 3605 y Fw(+)p Fn(`)2712 3620 y Fr(2)2752 3605 y Fw(\))2790 3569 y Fq(\000)p Fr(1)2884 3605 y Fw(+)17 b Fm(O)s Fw(\()p Fn(r)3144 3569 y Fq(\000)p Fr(1)3238 3605 y Fw(\))g(,)30 b(where)47 b Fn(r)409 3725 y Fw(is)33 b(the)g(gap)g(n)m(um)m(b)s(er.)165 3929 y(Most)h(of)e(these) i(results)g(ha)m(v)m(e)g(a)f(natural)g(meaning.)44 b(In)33 b(particular,)h(\(a5\))e(sho)m(ws)i(that)f(trans-)165 4049 y(p)s(ort)i(prop)s(erties)h(of)e(the)i(lattice)g(are)f(b)s(etter)g (than)g(a)g(com)m(bination)h(of)e(its)i(one{dimensional)165 4170 y(pro)5 b(jections.)44 b(Notice)28 b(that)g(the)h(Kronig{P)m (enney)g(sp)s(ectral)g(condition)2812 4133 y Fr(2)2881 4170 y Fw(can)f(b)s(e)g(cast)h(in)m(to)f(the)165 4290 y(form)g(\(Re)16 b(fdelta)29 b(band)g(condition\))g(with)g(a)f(single)h (trigonometric)g(expression)h(on)f(the)f Fo(lhs.)42 b Fw(If)165 4410 y(an)31 b(energy)h(v)-5 b(alue)31 b(is)h(con)m(tained)g (in)f(a)f(Kronig{P)m(enney)j(band)e(in)g(one)g(of)f(the)i(directions,)g (it)f(is)165 4531 y(trivially)i(also)f(in)g(a)g(band)g(of)f(the)h (lattice)h(Hamiltonian,)f(the)g(other)g(factor)g(b)s(eing)g(ann)m (ulated)165 4651 y(b)m(y)39 b(c)m(ho)s(osing)55 b Fn(\022)769 4666 y Fv(j)842 4651 y Fw(=)36 b Fn(k)20 b Fw(.)58 b(The)39 b(con)m(v)m(erse)h(is)e(not)f(true,)i(of)e(course:)55 b(the)38 b(condition)g(\(Re)16 b(fdelta)165 4772 y(band)39 b(condition\))h(ma)m(y)f(b)s(e)g(satis\014ed)i(ev)m(en)f(none)f(of)g (the)g(factors)g(can)g(b)s(e)g(ann)m(ulated)g(sepa-)165 4892 y(rately)-8 b(.)45 b(The)34 b(directions)h(in)e(whic)m(h)h(the)g (electron)g(is)f(able)h(to)e(\\dribble")i(through)f(the)g(lattice)165 5012 y(will)h(b)s(e)e(seen)i(in)f(Section)h(VI)e(b)s(elo)m(w.)331 5133 y(Less)41 b(trivial)g(is)g(the)f(irregular)g(dep)s(endence)j(of)d (the)h(sp)s(ectrum)g(on)f(the)h(rectangle{side)165 5253 y(ratio)62 b Fn(\022)52 b Fw(:=)e Fn(`)719 5268 y Fr(2)758 5253 y Fn(=`)848 5268 y Fr(1)949 5253 y Fw(coming)c(from)f(the)h (existence)h(of)e(comp)s(eting)h(p)s(erio)s(ds)g(in)62 b Fn(F)3321 5268 y Fq(\006)3380 5253 y Fw(\()p Fn(k)s Fw(\))17 b(.)82 b(It)165 5373 y(app)s(ears)35 b(that)e(it)h(is)g(not)g (only)h(rationalit)m(y)f(or)f(irrationalit)m(y)h(of)50 b Fn(\022)k Fw(whic)m(h)35 b(matters,)g(but)g(also)1913 5682 y(5)p eop %%Page: 6 6 6 5 bop 165 218 a Fw(the)35 b(t)m(yp)s(e)f(of)g(irrationalit)m(y)g(pla) m(ys)h(a)f(role.)48 b(In)34 b(this)g(resp)s(ect,)i(the)e(situation)h (is)f(similar)h(to)e(the)165 338 y(almost)g(Mathieu)h(equation)f(men)m (tioned)h(in)f(the)g(in)m(tro)s(duction.)331 459 y(Let)45 b(us)g(recall)h(some)f(elemen)m(tary)i(facts)e(from)g(the)g(n)m(um)m(b) s(er)i(theory)-8 b(.)3021 422 y Fr(20)3141 459 y Fw(An)45 b(irrational)165 579 y(n)m(um)m(b)s(er)66 b Fn(\022)h Fw(is)49 b Fo(b)-5 b(ad)5 b(ly)48 b(appr)-5 b(oximable)46 b Fw(b)m(y)j(rationals)e(if)h(there)g(is)g(a)g(p)s(ositiv)m(e)65 b Fn(\016)k Fw(suc)m(h)49 b(that)165 699 y Fm(j)p Fn(q)t(\022)-8 b Fm(\000)d Fn(p)p Fm(j)28 b Fn(>)f(\016)t(q)645 663 y Fq(\000)p Fr(1)764 699 y Fw(holds)e(for)f(all)h(in)m(tegers)42 b Fn(p;)17 b(q)j Fw(.)41 b(There)26 b(are)e(uncoun)m(tably)i(man)m(y)g (suc)m(h)g(n)m(um)m(b)s(ers;)165 820 y(nev)m(ertheless,)33 b(they)28 b(are)g(rather)g(exceptional)h(in)f(the)g(sense)h(that)f (they)g(form)g(a)f(zero{measure)165 940 y(set.)41 b(Its)23 b(complemen)m(t)h(to)e(the)g(set)h(of)f(all)g(irrationals)g(consists)i (of)e(n)m(um)m(b)s(ers)i(whic)m(h)f(w)m(e)h(shall)e(call)165 1060 y Fo(L)-5 b(ast)40 b(admissible.)848 1024 y Fr(5)923 1060 y Fw(A)e(con)m(v)m(enien)m(t)i(w)m(a)m(y)f(to)e(c)m(haracterize)i (these)g(n)m(um)m(b)s(er)g(t)m(yp)s(es)g(is)f(through)165 1181 y(their)46 b(unique)g(con)m(tin)m(ued{fraction)h(represen)m (tations:)87 b Fn(\022)65 b Fw(is)45 b(badly)h(appro)m(ximable)h(if)d (and)165 1301 y(only)32 b(if)e(the)h(in\014nite)h(sequence)h(of)d(in)m (teger)i(co)s(e\016cien)m(ts)h(in)e(this)g(represen)m(tation)i(is)e(b)s (ounded,)165 1422 y(and)c(Last)g(admissible)h(otherwise.)43 b(Needless)29 b(to)d(sa)m(y)-8 b(,)29 b(rationals)e(ha)m(v)m(e)h (\014nitely)g(man)m(y)f(nonzero)165 1542 y(co)s(e\016cien)m(ts.)331 1662 y(If)49 b Fn(\022)i Fw(is)33 b(irrational,)f(the)h Fo(rhs)66 b Fw(of)32 b(\(Re)16 b(fdelta)33 b(gap\))f(is)h(nev)m(er)g (zero;)g(the)g(existence)i(of)c(gaps)165 1783 y(requires)41 b(then)e(that)g(there)g(is)h(a)e(subsequence)k(of)d(lo)s(cal)f(minima)i (whic)m(h)g(tends)g(su\016cien)m(tly)165 1903 y(fast)33 b(to)f(zero.)44 b(In)33 b(this)g(w)m(a)m(y)h(w)m(e)g(w)m(ere)f(able)g (in)g(Ref.13)f(to)h(pro)m(v)m(e)g(the)g(follo)m(wing)g(results:)214 2106 y(\(a7\))43 b(F)-8 b(or)43 b(a)g(badly)h(appro)m(ximable)62 b Fn(\022)h Fw(there)44 b(is)61 b Fn(\013)2179 2121 y Fr(0)2264 2106 y Fn(>)47 b Fw(0)59 b(suc)m(h)45 b(that)f(for)59 b Fm(j)p Fn(\013)q Fm(j)45 b Fn(<)i(\013)3474 2121 y Fr(0)3573 2106 y Fw(the)409 2227 y(sp)s(ectrum)34 b(has)f(no)g(gaps)f (ab)s(o)m(v)m(e)i(the)f(threshold.)214 2353 y(\(a8\))c(The)i(n)m(um)m (b)s(er)g(of)e(gaps)h(is)g(in\014nite)h(for)e(an)m(y)47 b Fn(\022)i Fw(pro)m(vided)f Fm(j)p Fn(\013)q Fm(j)p Fn(L)27 b(>)g Fw(5)2935 2317 y Fq(\000)p Fr(1)p Fv(=)p Fr(2)3100 2353 y Fn(\031)3159 2317 y Fr(2)3215 2353 y Fw(;)k(recall)f(that)409 2473 y Fn(L)e Fw(:=)g(max\()p Fn(`)894 2488 y Fr(1)934 2473 y Fn(;)17 b(`)1019 2488 y Fr(2)1058 2473 y Fw(\))g(.)214 2600 y(\(a9\))44 b(If)61 b Fn(\022)j Fw(is)45 b(rational)f(or)g(Last)g(admissible,)50 b(there)45 b(are)f(in\014nitely)i(man)m(y)g(gaps)e(for)g(an)m(y)409 2720 y Fn(\013)29 b Fm(6)p Fw(=)e(0)17 b(.)331 2923 y(The)46 b(w)m(orst)h(irrational)e(in)h(this)g(sense)i(is)e(the)g(golden)g(mean) 63 b Fn(\022)53 b Fw(=)2993 2884 y Fr(1)p 2993 2900 36 4 v 2993 2958 a(2)3039 2923 y Fw(\(1)14 b(+)3230 2841 y Fm(p)p 3313 2841 49 4 v 82 x Fw(5\))62 b(whic)m(h)165 3044 y(has)36 b(the)g(con)m(tin)m(ued{fraction)h(represen)m(tation)54 b Fn(\022)36 b Fw(=)d([1)p Fn(;)17 b Fw(1)p Fn(;)g(:)g(:)g(:)n Fw(])g(.)52 b(In)36 b(this)g(case)h(the)e(su\016cien)m(t)165 3164 y(condition)29 b(for)e(the)h(existence)j(of)c(in\014nitely)i(man)m (y)g(gaps)f(is)h(necessary)h(at)d(the)h(same)h(time)f(and)165 3285 y(coincides)35 b(with)e(the)g(critical)h(v)-5 b(alue)33 b(of)f(the)h(claim)h(\(a7\):)59 b Fm(j)p Fn(\013)2436 3300 y Fr(0)2475 3285 y Fm(j)p Fn(L)50 b Fw(is)g Fn(\031)2793 3248 y Fr(2)2832 3285 y Fw(\(5)p Fn(\022)s Fw(\))3005 3248 y Fq(\000)p Fr(1)p Fv(=)p Fr(2)3198 3285 y Fw(=)27 b(3)p Fn(:)p Fw(4699)p Fn(:::)h Fw(;)165 3405 y(more)33 b(ab)s(out)f(that)h(will)g(b)s(e)g(said)g(in)g(Section)g(V)g(b)s(elo)m (w.)165 3694 y Fx(I)t(I)t(I.2)158 b Fk(\016)652 3650 y Fi(0)647 3723 y Fh(s)756 3694 y Fx(lattices)165 3879 y Fw(Replacing)28 b(the)44 b Fn(\016)j Fw(coupling)27 b(b)m(y)h(the)f(b)s(oundary)g(conditions)h(\(Re)16 b(fdelta')28 b(b)s(c\),)g(one)f(can)g(deriv)m(e)165 3999 y(the)33 b(band)g(equation)h(in)e(this)i(case:)1480 3963 y Fr(13)845 4195 y Fw(cos)17 b Fn(#)1049 4210 y Fr(1)1089 4195 y Fn(`)1130 4210 y Fr(1)1192 4195 y Fw(+)22 b(cos)17 b Fn(k)s(`)1532 4210 y Fr(1)p 845 4240 727 4 v 1073 4331 a Fw(sin)g Fn(k)s(`)1304 4346 y Fr(1)1620 4263 y Fw(+)1745 4195 y(cos)g Fn(#)1949 4210 y Fr(2)1989 4195 y Fn(`)2030 4210 y Fr(2)2092 4195 y Fw(+)22 b(cos)17 b Fn(k)s(`)2432 4210 y Fr(2)p 1745 4240 V 1973 4331 a Fw(sin)g Fn(k)s(`)2204 4346 y Fr(2)2520 4263 y Fm(\000)2646 4195 y Fn(\014)6 b(k)p 2646 4240 115 4 v 2679 4331 a Fw(2)2815 4263 y(=)44 b(0)28 b(;)545 b(\(5\))165 4527 y(the)36 b(same)g(argumen)m(t)g(as)g (ab)s(o)m(v)m(e)g(then)g(sho)m(ws)h(that)e(sp)s(ectral)h Fo(b)-5 b(ands)35 b Fw(of)g(the)52 b Fn(\016)3133 4491 y Fq(0)3129 4552 y Fv(s)3218 4527 y Fw(lattice)36 b(with)165 4647 y(a)d(coupling)g(constan)m(t)50 b Fn(\014)55 b Fw(are)32 b(determined)j(b)m(y)e(the)g(inequalities)1575 4911 y Fm(\007)22 b Fn(F)1737 4926 y Fq(\007)1797 4911 y Fw(\()p Fn(k)s Fw(\))44 b Fm(\025)h(\006)2197 4844 y Fn(\014)6 b(k)p 2197 4888 V 2230 4980 a Fw(2)3584 4911 y(\(6\))165 5157 y(for)49 b Fm(\006)p Fn(\014)34 b(>)27 b Fw(0)49 b(and)g Fn(k)31 b(>)d Fw(0)17 b(,)32 b(and)g(an)h(analogous)f(relation) h(for)f(the)h(negativ)m(e)h(part.)331 5277 y(The)c(structure)g(of)f (the)g(sp)s(ectrum)i(is)e(no)m(w)h(di\013eren)m(t;)i(the)d(condition)h (\(6\))f(allo)m(ws)h(to)e(mak)m(e)165 5398 y(the)33 b(follo)m(wing)g (conclusions:)1913 5682 y(6)p eop %%Page: 7 7 7 6 bop 214 218 a Fw(\(b1\))24 b(The)g(sp)s(ectrum)i(equals)41 b([0)p Fn(;)17 b Fm(1)p Fw(\))40 b(if)23 b(and)h(only)g(if)41 b Fn(\014)33 b Fw(=)27 b(0)h(;)f(otherwise)e(there)f(are)g (in\014nitely)409 338 y(man)m(y)34 b(gaps.)214 464 y(\(b2\))h(If)52 b Fn(\014)38 b(>)32 b Fw(0)17 b(,)35 b(the)h(lo)m(w)m(er)g(end)g(of)e (eac)m(h)j(band)e(coincides)i(with)f(some)53 b Fn(k)3029 428 y Fr(2)3026 489 y Fv(n)3124 464 y Fw(or)3265 438 y(~)3263 464 y Fn(k)3317 428 y Fr(2)3314 489 y Fv(m)3396 464 y Fw(,)36 b(where)409 585 y Fn(n;)e(m)46 b Fw(are)30 b(in)m(tegers.)44 b(The)30 b(same)h(is)f(true)g(for)45 b Fn(\014)33 b(<)28 b Fw(0)46 b(and)30 b(the)g(upp)s(er)g(band)g(ends,) h(with)409 705 y(the)i(exception)h(of)f(the)g(\014rst)g(one.)214 831 y(\(b3\))38 b(The)g(lo)m(w)m(est)i(band)e(threshold)g(is)h(p)s (ositiv)m(e)g(for)53 b Fn(\014)42 b(>)37 b Fw(0)54 b(and)38 b(negativ)m(e)g(if)55 b Fn(\014)42 b(<)36 b Fw(0)g(;)k(in)409 952 y(the)e(case)54 b Fm(\000)p Fn(`)927 967 y Fr(1)992 952 y Fm(\000)26 b Fn(`)1136 967 y Fr(2)1211 952 y Fn(<)35 b(\014)41 b(<)35 b Fw(0)54 b(the)37 b(whole)h(\014rst)g(band)f(is)h (negativ)m(e,)i(and)d(the)g(second)409 1072 y(one)c(starts)g(at)f (zero.)214 1198 y(\(b4\))g(The)i(p)s(ositiv)m(e)g(bands)f(shrink)h (with)f(increasing)51 b Fm(j)p Fn(\014)6 b Fm(j)17 b Fw(.)214 1324 y(\(b5\))29 b(Eac)m(h)i(gap)e(is)h(con)m(tained)h(in)f (the)g(in)m(tersection)h(of)e(a)g(pair)h(of)46 b Fn(\016)2729 1288 y Fq(0)2798 1324 y Fw(Kronig{P)m(enney)31 b(gaps)3669 1288 y Fr(2)409 1445 y Fw(with)i(the)g(coupling)h(constan)m(t)50 b Fn(\014)k Fw(and)33 b(spacings)51 b Fn(`)2343 1460 y Fr(1)2431 1445 y Fw(and)e Fn(`)2678 1460 y Fr(2)2734 1445 y Fw(,)33 b(resp)s(ectiv)m(ely)-8 b(.)331 1648 y(Instead)46 b(of)e(the)i(asymptotic)g(b)s(eha)m(vior)f(\(a6\))g(of)61 b Fn(\016)t Fw({lattices,)49 b(w)m(e)d(ha)m(v)m(e)g(sligh)m(tly)g(more) 165 1769 y(complicated)35 b(result.)48 b(If)34 b(a)f(band)h(high)g(in)g (the)g(sp)s(ectrum)h(is)f(w)m(ell)h(separated,)g(its)f(width)51 b(\001)3654 1784 y Fv(r)165 1889 y Fw(is)37 b(the)f(same)g(as)g(in)g (the)53 b Fn(\016)1160 1853 y Fq(0)1219 1889 y Fw(Kronig{P)m(enney)37 b(mo)s(del,)54 b(\001)2325 1904 y Fv(r)2396 1889 y Fw(=)2549 1850 y Fr(8)p 2515 1866 105 4 v 2515 1923 a Fv(\014)s(`)2587 1933 y Fg(j)2654 1889 y Fw(+)24 b Fm(O)s Fw(\()p Fn(r)2921 1853 y Fq(\000)p Fr(1)3015 1889 y Fw(\))17 b(.)52 b(F)-8 b(urthermore,)165 2044 y(if)48 b Fn(\022)j Fw(is)31 b(rational)g(and)g (t)m(w)m(o)h(bands)g(in)m(tersect,)50 b Fn(k)1963 2059 y Fv(n)2037 2044 y Fw(=)2144 2018 y(~)2141 2044 y Fn(k)2192 2059 y Fv(m)2306 2044 y Fw(for)31 b(some)48 b Fn(n;)17 b(m)g Fw(,)31 b(w)m(e)i(ha)m(v)m(e)f(a)f(similar)165 2164 y(expression)39 b(with)54 b Fn(`)923 2123 y Fq(\000)p Fr(1)923 2187 y Fv(j)1054 2164 y Fw(replaced)38 b(b)m(y)54 b Fn(`)1640 2123 y Fq(\000)p Fr(1)1640 2186 y(1)1742 2164 y Fw(+)25 b Fn(`)1884 2123 y Fq(\000)p Fr(1)1884 2186 y(2)1978 2164 y Fw(.)56 b(It)37 b(ma)m(y)g(happ)s(en,)i(ho)m(w)m (ev)m(er,)h(that)53 b Fn(k)3452 2179 y Fv(n)3551 2164 y Fw(and)168 2274 y(~)165 2301 y Fn(k)216 2316 y Fv(m)344 2301 y Fw(are)44 b(not)h(iden)m(tical)h(but)e(close)i(to)e(eac)m(h)i (other,)h(so)e(that)g(they)g(still)g(pro)s(duce)g(a)g(single)165 2421 y(band.)e(Then)31 b(the)g(band)f(width)h(is)g(enhanced;)h(the)f (e\013ect)g(is)f(most)h(profound)f(just)g(b)s(efore)g(the)165 2541 y(band)39 b(splits.)61 b(Using)39 b(the)g(condition)f(\(5\),)i(it) e(is)g(straigtforw)m(ard)h(to)e(estimate)j(the)e(factor)g(of)165 2662 y(enhancemen)m(t)d(b)m(y)f(conspiracy)g(of)e(bands:)1756 2626 y Fr(13)1864 2662 y Fw(asymptotically)j(w)m(e)e(ha)m(v)m(e)h(the)f (relation)925 2853 y(8)p 886 2898 127 4 v 886 2989 a Fn(\014)6 b(L)1061 2921 y Fw(+)39 b Fm(O)s Fw(\()p Fn(r)1343 2880 y Fq(\000)p Fr(1)1437 2921 y Fw(\))44 b Fn(<)g Fw(\001)1720 2936 y Fv(r)1803 2921 y Fn(<)1939 2853 y Fw(32)p 1933 2898 110 4 v 1933 2989 a(3)p Fn(\014)2069 2921 y Fw(\()p Fn(`)2148 2880 y Fq(\000)p Fr(1)2148 2945 y(1)2248 2921 y Fw(+)6 b Fn(`)2371 2880 y Fq(\000)p Fr(1)2371 2945 y(2)2464 2921 y Fw(\))39 b(+)g Fm(O)s Fw(\()p Fn(r)2823 2880 y Fq(\000)p Fr(1)2917 2921 y Fw(\))17 b Fn(:)585 b Fw(\(7\))331 3186 y(This)36 b(brief)g(surv)m(ey)h(sho)m(ws,)h(in)e (particular,)g(that)f(despite)i(the)f(generally)g(irregular)f(pat-)165 3306 y(tern,)28 b(the)f(sp)s(ectra)g(of)42 b Fn(\016)47 b Fw(and)c Fn(\016)1346 3270 y Fq(0)1342 3331 y Fv(s)1421 3306 y Fw(lattices)27 b(inherit)g(the)g(main)f(feature)h(of)f(their)g (Kronig{P)m(eney)165 3426 y(analogues,)37 b(namely)f(that)f(they)h(are) f(dominated)h(asymptotically)h(b)m(y)f(bands)g(and)f(gaps,)h(re-)165 3547 y(sp)s(ectiv)m(ely)-8 b(,)39 b(at)c(high)g(energies.)52 b(A)m(t)36 b(the)f(same)h(time,)h(the)e(ab)s(o)m(v)m(e)h(listed)g (results)h(lea)m(v)m(e)g(man)m(y)165 3667 y(questions)31 b(op)s(en)e(ab)s(out)f(the)h(actual)f(form)g(of)h(these)g(sp)s(ectra)h (and)e(related)h(quan)m(tities.)44 b(In)29 b(the)165 3788 y(follo)m(wing)k(sections)h(w)m(e)g(are)f(going)f(to)g(answ)m(er)i (some)f(of)f(them.)165 4120 y Fp(IV)189 b Ff(\016)84 b Fp(lattice)55 b(sp)5 b(ectra)165 4339 y Fw(The)41 b(form)e(\(4\))h (of)f(the)h(gap)f(condition)h(mak)m(es)i(it)d(p)s(ossible)i(to)f (\014nd)g(the)57 b Fn(\016)j Fw(lattice)40 b(sp)s(ectra)165 4460 y(n)m(umerically)j(in)f(dep)s(endence)h(on)e(parameters)h(of)e (the)i(mo)s(del;)k(the)41 b(results)h(are)f(sho)m(wn)i(on)165 4580 y(Figures)35 b(2)f(and)g(3.)47 b(The)35 b(e\013ect)g(of)f(comp)s (eting)g(p)s(erio)s(ds)h(is)f(ob)m(vious:)48 b(while)35 b(a)f(square)h(lattice)165 4700 y(has)k(a)e(familiar)h(Kronig{P)m (enney)i(sp)s(ectrum)f(shap)s(e)2159 4664 y Fr(2)2237 4700 y Fw(with)g(the)f(halv)m(ed)h(coupling)g(constan)m(t)165 4821 y(as)31 b(exp)s(ected)h(from)e(the)h(gap)e(condition,)j(in)e(the)h (general)g(case)g(the)f(gap)g(pattern)h(is)f(irregular.)331 4941 y(The)40 b(dep)s(endence)h(of)d(sp)s(ectral)i(bands)g(on)f(the)g (ratio)55 b Fn(\022)k Fw(sho)m(ws)40 b(clearly)g(that)f(the)g(gaps)165 5062 y(are)i(con)m(tained)h(in)f(view)h(of)f(\(a5\))f(in)h(the)g(in)m (tersections)i(of)d(KP{gaps.)69 b(The)42 b(latter)e(can)h(b)s(e)165 5182 y(n)m(um)m(b)s(ered)26 b(b)m(y)f(an)e(in)m(teger)41 b Fn(n)g Fw(suc)m(h)25 b(that)40 b Fn(\031)t(n`)1879 5141 y Fq(\000)p Fr(1)1879 5205 y Fv(j)1997 5182 y Fw(is)24 b(the)g(lo)m(w)m(er)h(\(upp)s(er\))f(gap)g(end)g(for)40 b Fm(\006)p Fn(\013)29 b(>)e Fw(0)17 b(,)165 5302 y(resp)s(ectiv)m(ely) 28 b(\(with)43 b Fn(n)28 b Fw(=)f(0)41 b(referring)26 b(to)e(the)i(region)f(b)s(elo)m(w)h(the)f(b)s(ottom)g(of)f(the)i(sp)s (ectrum)3603 5266 y Fr(2)3644 5302 y Fw(\).)165 5423 y(This)33 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Fw(Figure)30 b(4a:)41 b(The)31 b(folded)46 b Fn(\016)k Fw(lattice)30 b(sp)s(ectrum)h(as)e(a)g(function)h(of)46 b(ln)17 b Fn(\022)j Fw(:)42 b(The)30 b(regions)g(with)g(at)165 4864 y(least)k(one)e(gap;)h(the)g(lo)m(w)m(est)h(one,)f(\(1,1\),)f(is)h (deleted.)1888 5682 y(11)p eop %%Page: 12 12 12 11 bop 165 4626 a @beginspecial -5 @llx 93 @lly 449 @urx 531 @ury 4540 @rwi @setspecial %%BeginDocument: Bild4_2.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def /rgbindx 0 def -7 25 translate 612.00000 792.00000 scale 612 792 1 [612 0 0 -792 0 792] {currentfile pix readhexstring pop} image ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 120 x Fw(Figure)41 b(4c:)60 b(The)42 b(folded)57 b Fn(\016)k Fw(lattice)42 b(sp)s(ectrum)g(as)f(a)f(function)h(of)57 b(ln)17 b Fn(\022)j Fw(:)59 b(The)42 b(plot)f(of)f(gap)165 4901 y(edges)34 b(in)f(the)g(same)g(scale)h(as)f(\(a\).)1888 5682 y(13)p eop %%Page: 14 14 14 13 bop 165 4563 a @beginspecial -5 @llx 93 @lly 453 @urx 531 @ury 4580 @rwi @setspecial %%BeginDocument: Bild4_4.epsi /InitDictCount 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9.136881E+02 a 2.342046E+02 3.470091E+01 a 2.365784E+02 6.995223E+02 a 2.379795E+02 6.214130E+02 a 2.401256E+02 3.127487E+02 a 2.434153E+02 2.247455E+02 a 2.453354E+02 6.624555E+02 a 2.482561E+02 1.735552E+02 a 2.510129E+02 6.196809E+02 a 2.526195E+02 5.565533E+02 a 2.551034E+02 5.614726E+01 a 2.588556E+02 3.820351E+02 a 2.610244E+02 5.060380E+02 a 2.623105E+02 8.707339E+02 a 2.643076E+02 1.388123E+02 a 2.673689E+02 6.604313E+02 a 2.691549E+02 2.808183E+02 a 2.718992E+02 3.818803E+02 a 2.735183E+02 9.005221E+02 a 2.760012E+02 3.469955E+01 a 2.783591E+02 6.947879E+02 a 2.797543E+02 6.181459E+02 a 2.819167E+02 3.123601E+02 a 2.852064E+02 2.246030E+02 a 2.871223E+02 6.597472E+02 a 2.900512E+02 1.735051E+02 a 2.927980E+02 6.178953E+02 a 2.944106E+02 5.553876E+02 a 2.969000E+02 5.614505E+01 a 3.006498E+02 3.817424E+02 a 3.028155E+02 5.054093E+02 a 3.041016E+02 8.679555E+02 a 3.061042E+02 1.387995E+02 a 3.091626E+02 6.594537E+02 a 3.109500E+02 2.807372E+02 a 3.136949E+02 3.817199E+02 a 3.153094E+02 8.986361E+02 a 3.177986E+02 3.469935E+01 a 3.201542E+02 6.941032E+02 a 3.215486E+02 6.176722E+02 a 3.237133E+02 3.123035E+02 a 3.270030E+02 2.245822E+02 a 3.289182E+02 6.593541E+02 a 3.318484E+02 1.734978E+02 a 3.345937E+02 6.176357E+02 a 3.362072E+02 5.552180E+02 a 3.386974E+02 5.614473E+01 a 3.424468E+02 3.816998E+02 a 3.446121E+02 5.053177E+02 a 3.458982E+02 8.675518E+02 a 3.479016E+02 1.387976E+02 a 3.509596E+02 6.593113E+02 a 3.527471E+02 2.807254E+02 a 3.554922E+02 3.816965E+02 a 3.571060E+02 8.983616E+02 a 3.595961E+02 3.469932E+01 a 3.619514E+02 6.940034E+02 a 3.633456E+02 6.176032E+02 a 3.655107E+02 3.122953E+02 a 3.688004E+02 2.245792E+02 a 3.707155E+02 6.592968E+02 a 3.736458E+02 1.734968E+02 a 3.763909E+02 6.175979E+02 a 3.780046E+02 5.551933E+02 a 3.804949E+02 5.614468E+01 a 3.842443E+02 3.816935E+02 a 3.864095E+02 5.053043E+02 a 3.876956E+02 8.674930E+02 a 3.896991E+02 1.387973E+02 a 3.927570E+02 6.592906E+02 a 3.945446E+02 2.807237E+02 a 3.972897E+02 3.816931E+02 a 3.989034E+02 8.983216E+02 a 4.013936E+02 3.469932E+01 a 4.037488E+02 6.939888E+02 a 4.051431E+02 6.175931E+02 a 4.073082E+02 3.122941E+02 a 4.105979E+02 2.245788E+02 a 4.125130E+02 6.592884E+02 a 4.154434E+02 1.734966E+02 a 4.181884E+02 6.175924E+02 a 4.198021E+02 5.551897E+02 a 4.222924E+02 5.614468E+01 a 4.260418E+02 3.816926E+02 a 4.282070E+02 5.053024E+02 a 4.294931E+02 8.674844E+02 a 4.314966E+02 1.387973E+02 a 4.345545E+02 6.592875E+02 a 4.363421E+02 2.807234E+02 a 4.390872E+02 3.816926E+02 a 4.407009E+02 8.983158E+02 a 4.431911E+02 3.469932E+01 a 4.455464E+02 6.939867E+02 a 4.469406E+02 6.175917E+02 a 4.491057E+02 3.122939E+02 a 4.523954E+02 2.245787E+02 a 4.543105E+02 6.592872E+02 a 4.572409E+02 1.734966E+02 a 4.599859E+02 6.175916E+02 a 4.615996E+02 5.551892E+02 a 4.640899E+02 5.614467E+01 a 4.678393E+02 3.816925E+02 a 4.700045E+02 5.053021E+02 a 4.712906E+02 8.674831E+02 a 4.732941E+02 1.387973E+02 a 4.763521E+02 6.592871E+02 a 4.781396E+02 2.807234E+02 a 4.808847E+02 3.816925E+02 a 4.824984E+02 8.983149E+02 a 4.849887E+02 3.469931E+01 a 4.873439E+02 6.939864E+02 a 4.887381E+02 6.175915E+02 a 4.909033E+02 3.122939E+02 a 4.941929E+02 2.245787E+02 a 4.961081E+02 6.592870E+02 a 4.990384E+02 1.734966E+02 a 5.017835E+02 6.175914E+02 a 5.033971E+02 5.551891E+02 a 5.058874E+02 5.614468E+01 a 5.096369E+02 3.816925E+02 a 5.118020E+02 5.053021E+02 a 5.130882E+02 8.674829E+02 a 5.150917E+02 1.387973E+02 a 5.181496E+02 6.592870E+02 a 5.199372E+02 2.807234E+02 a 5.226822E+02 3.816925E+02 a 5.242959E+02 8.983148E+02 a 5.267862E+02 3.469929E+01 a 5.291414E+02 6.939863E+02 a 5.305356E+02 6.175914E+02 a 5.327008E+02 3.122938E+02 a 5.359904E+02 2.245787E+02 a 5.379056E+02 6.592870E+02 a 5.408359E+02 1.734965E+02 a 5.435810E+02 6.175914E+02 a 5.451947E+02 5.551890E+02 a 5.476850E+02 5.614471E+01 a 5.514344E+02 3.816924E+02 a 5.535996E+02 5.053021E+02 a 5.548857E+02 8.674828E+02 a 5.568892E+02 1.387972E+02 a 5.599471E+02 6.592869E+02 a 5.617347E+02 2.807234E+02 a 5.644798E+02 3.816924E+02 a 5.660934E+02 8.983149E+02 a 5.685837E+02 3.469917E+01 a 5.709389E+02 6.939862E+02 a 5.723332E+02 6.175915E+02 a 5.744983E+02 3.122937E+02 a 5.777880E+02 2.245788E+02 a 5.797031E+02 6.592868E+02 a 5.826335E+02 1.734963E+02 a 5.853785E+02 6.175916E+02 a 5.869922E+02 5.551887E+02 a 5.894825E+02 5.614494E+01 a 5.932319E+02 3.816920E+02 a 5.953971E+02 5.053024E+02 a 5.966832E+02 8.674824E+02 a 5.986867E+02 1.387967E+02 a showpage grestore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 359 x Fw(Figure)33 b(5:)43 b(Critical)33 b(coupling)h(constan)m(ts)g(for)e(the)h(golden{mean)49 b Fn(\016)54 b Fw(lattice.)1888 5682 y(15)p eop %%Page: 16 16 16 15 bop 409 218 a Fw(yields)42 b(a)e(critical)h(v)-5 b(alue:)60 b(in)40 b(b)s(oth)g(series)i(the)f(same)g(sequence)i(of)d (in)m(tegers)i(rep)s(eats,)409 338 y(namely)1389 459 y(1)p Fn(;)33 b Fw(4)p Fn(;)g Fw(5)p Fn(;)g Fw(9)p Fn(;)g Fw(11)p Fn(;)g Fw(16)p Fn(;)g Fw(19)p Fn(;)g Fw(20)p Fn(;)g Fw(25)p Fn(;)f(:)17 b(:)g(:)855 b Fw(\(8\))165 643 y(Let)41 b(us)g(attempt)g(to)f(explain)h(this)g(quan)m(tization)g (rule.)68 b(Recall)41 b(that)f(the)g(golden)h(mean)g(is)165 763 y(appro)m(ximated)g(b)m(y)f(ratios)f(of)g(successiv)m(e)k(Fib)s (onacci)d(n)m(um)m(b)s(ers,)2625 727 y Fr(17)2780 763 y Fn(\022)i Fw(=)d(lim)3118 778 y Fv(n)p Fq(!1)3322 763 y Fn(u)3378 778 y Fv(n)p Fr(+1)3515 763 y Fn(u)3571 727 y Fq(\000)p Fr(1)3571 788 y Fv(n)3682 763 y Fw(,)165 884 y(where)30 b(the)45 b Fn(u)679 899 y Fv(n)770 884 y Fw(satisfy)28 b(the)h(recursiv)m(e)h(relation)45 b Fn(u)2061 899 y Fv(n)p Fr(+1)2225 884 y Fw(=)28 b Fn(u)2385 899 y Fv(n)2444 884 y Fw(+)13 b Fn(u)2589 899 y Fv(n)p Fq(\000)p Fr(1)2770 884 y Fw(and)28 b(assume)h(the)g(v)-5 b(alues)165 1004 y(1)p Fn(;)17 b Fw(2)p Fn(;)g Fw(3)p Fn(;)g Fw(5)p Fn(;)g Fw(8)p Fn(;)g Fw(13)p Fn(;)g Fw(21)p Fn(;)g(:)g(:)g(:)46 b Fw(for)j Fn(n)29 b Fw(=)g(1)p Fn(;)17 b Fw(2)p Fn(;)g(:)g(:)g(:)e Fw(.)46 b(In)34 b(the)f(con)m(tin)m (ued{fraction)i(represen)m(tation)g(the)165 1125 y Fn(n)p Fw({th)e(appro)m(ximan)m(t)h(equals)50 b Fn(\022)1326 1140 y Fv(n)1385 1125 y Fw(=)27 b([1)p Fn(;)17 b(:)g(:)g(:)33 b Fw(1)p Fn(;)17 b Fw(0)p Fn(;)g(:)g(:)g(:)n Fw(])49 b(with)h Fn(n)6 b Fw(+)g(1)48 b(nonzero)33 b(co)s(e\016cien)m(ts.)331 1245 y(Let)42 b(us)g(\\sp)s(oil")g(the)g(appro)m(ximation)g(b)m(y)h(c)m (ho)s(osing)g(instead)f(the)h(sequence)h(of)58 b Fn(\022)3495 1209 y Fv(N)3492 1270 y(n)3606 1245 y Fw(:=)165 1365 y([1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fw(1)p Fn(;)g(N)5 b(;)17 b Fw(0)p Fn(;)g(:)g(:)g(:)n Fw(])55 b(with)39 b(a)f(\014xed)h(p)s (ositiv)m(e)h(in)m(teger)55 b Fn(N)66 b Fw(preceded)40 b(b)m(y)56 b Fn(n)f Fw(ones.)61 b(This)39 b(can)165 1486 y(b)s(e)33 b(alternativ)m(ely)h(written)g(as)49 b Fn(q)1385 1501 y Fv(n)p Fr(+1)1523 1486 y Fn(q)1570 1450 y Fq(\000)p Fr(1)1566 1510 y Fv(n)1680 1486 y Fw(,)33 b(where)50 b Fn(q)2081 1501 y Fv(n)2156 1486 y Fw(:=)28 b Fn(N)10 b(u)2431 1501 y Fv(n)2484 1486 y Fw(+)22 b Fn(u)2638 1501 y Fv(n)p Fq(\000)p Fr(1)2791 1486 y Fw(,)32 b(so)h(that)253 1729 y Fn(q)300 1688 y Fr(2)296 1754 y Fv(n)360 1583 y Fl( )426 1729 y Fn(\022)25 b Fm(\000)622 1662 y Fn(q)665 1677 y Fv(n)p Fr(+1)p 622 1706 181 4 v 667 1798 a Fn(q)710 1813 y Fv(n)812 1583 y Fl(!)956 1729 y Fw(=)1111 1583 y Fl(")1176 1729 y Fn(N)1291 1662 y(\022)1339 1626 y Fv(n)1391 1662 y Fm(\000)e Fw(\()p Fm(\000)p Fn(\022)s Fw(\))1692 1626 y Fq(\000)p Fv(n)p 1291 1706 504 4 v 1477 1724 a Fm(p)p 1560 1724 49 4 v 82 x Fw(5)1843 1729 y(+)1968 1662 y Fn(\022)2016 1626 y Fv(n)p Fq(\000)p Fr(1)2158 1662 y Fm(\000)g Fw(\()p Fm(\000)p Fn(\022)s Fw(\))2459 1626 y Fq(\000)p Fv(n)p Fr(+1)p 1968 1706 685 4 v 2244 1724 a Fm(p)p 2327 1724 49 4 v 82 x Fw(5)2678 1583 y Fl(#)2760 1729 y Fw(\()p Fm(\000)p Fn(\022)s Fw(\))2961 1688 y Fq(\000)p Fv(n)p Fr(+1)3187 1608 y Fl(\022)3248 1729 y Fw(1)f Fm(\000)3445 1662 y Fn(N)p 3445 1706 89 4 v 3465 1798 a(\022)3543 1608 y Fl(\023)945 2054 y Fm(!)66 b Fw(\()p Fm(\000)p 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 120 x Fw(Figure)25 b(6c:)40 b(F)-8 b(ermi)25 b(surfaces)h(for)f(di\013eren)m(t)h(energies)g(for)f(the)g(second)h (band)g(of)e(a)h(golden{mean)165 5185 y Fn(\016)53 b Fw(lattice)1888 5682 y(20)p eop %%Page: 21 21 21 20 bop 225 4449 a @beginspecial 125 @llx 264 @lly 536 @urx 674 @ury 4110 @rwi @setspecial %%BeginDocument: Bild6_4.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def /rgbindx 0 def 0 0 translate 612.00000 792.00000 scale 612 792 1 [612 0 0 -792 0 792] {currentfile pix readhexstring pop} image 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L 3124 1469 L 3130 1473 L 3136 1477 L 3142 1481 L 3148 1485 L 3154 1490 L 3160 1494 L 3166 1498 L 3172 1502 L 3178 1506 L 3184 1511 L 3190 1515 L 3196 1519 L 3202 1523 L 3208 1528 L 3214 1532 L 3220 1536 L 3225 1540 L 3231 1545 L 3237 1549 L 3243 1553 L 3249 1557 L 3255 1561 L 3261 1566 L 3267 1570 L 3273 1574 L 3279 1579 L 3285 1583 L 3291 1587 L 3297 1591 L 3303 1596 L 3309 1600 L 3315 1604 L 3321 1608 L 3327 1613 L 3333 1617 L 3339 1621 L 3345 1626 L 3351 1630 L 3357 1634 L 3363 1639 L 3369 1643 L 3375 1647 L currentpoint stroke moveto 3380 1651 L 3386 1656 L 3392 1660 L 3398 1664 L 3404 1669 L 3410 1673 L 3416 1677 L 3422 1682 L 3428 1686 L 3434 1690 L 3440 1695 L 3446 1699 L 3452 1703 L 3458 1708 L 3464 1712 L 3470 1716 L 3476 1721 L 3482 1725 L 3488 1729 L 3494 1734 L 3500 1738 L 3506 1742 L 3512 1747 L 3518 1751 L 3524 1756 L 3530 1760 L 3535 1764 L 3541 1769 L 3547 1773 L 3553 1777 L 3559 1782 L 3565 1786 L 3571 1791 L 3577 1795 L 3583 1799 L 3589 1804 L 3595 1808 L 3601 1813 L 3607 1817 L 3613 1821 L 3619 1826 L 3625 1830 L 3631 1834 L 3637 1839 L 3643 1843 L 3649 1848 L 3655 1852 L 3661 1857 L 3667 1861 L 3673 1865 L 3679 1870 L 3684 1874 L 3690 1879 L 3696 1883 L 3702 1887 L 3708 1892 L 3714 1896 L 3720 1901 L 3726 1905 L 3732 1910 L 3738 1914 L 3744 1918 L 3750 1923 L 3756 1927 L 3762 1932 L 3768 1936 L 3774 1941 L 3780 1945 L 3786 1949 L 3792 1954 L 3798 1958 L 3804 1963 L 3810 1967 L 3816 1972 L 3822 1976 L 3828 1981 L 3834 1985 L 3839 1990 L 3845 1994 L 3851 1998 L 3857 2003 L 3863 2007 L 3869 2012 L 3875 2016 L 3881 2021 L 3887 2025 L 3893 2030 L 3899 2034 L 3905 2039 L 3911 2043 L 3917 2048 L 3923 2052 L 3929 2056 L 3935 2061 L 3941 2065 L 3947 2070 L 3953 2074 L 3959 2079 L 3965 2083 L 3971 2088 L 3977 2092 L 3983 2097 L 3989 2101 L 3994 2106 L 4000 2110 L 4006 2115 L 4012 2119 L 4018 2124 L 4024 2128 L 4030 2133 L 4036 2137 L 4042 2142 L 4048 2146 L 4054 2151 L 4060 2155 L 4066 2160 L 4072 2164 L 4078 2169 L 4084 2173 L 4090 2178 L 4096 2182 L 4102 2187 L 4108 2191 L 4114 2196 L 4120 2200 L 4126 2205 L 4132 2209 L 4138 2214 L 4143 2218 L 4149 2223 L 4155 2227 L 4161 2232 L 4167 2236 L 4173 2241 L 4179 2245 L 4185 2250 L 4191 2254 L 4197 2259 L 4203 2263 L 4209 2268 L 4215 2272 L 4221 2277 L 4227 2281 L 4233 2286 L 4239 2290 L 4245 2295 L 4251 2299 L 4257 2304 L 4263 2308 L 4269 2313 L 4275 2317 L 4281 2322 L 4287 2326 L 4293 2331 L 4298 2335 L 4304 2340 L 4310 2344 L 4316 2349 L 4322 2353 L 4328 2358 L 4334 2363 L 4340 2367 L 4346 2372 L 4352 2376 L 4358 2381 L 4364 2385 L 4370 2390 L 4376 2394 L 4382 2399 L 4388 2403 L 4394 2408 L 4400 2412 L 4406 2417 L 4412 2421 L 4418 2426 L 4424 2430 L 4430 2435 L 4436 2439 L 4442 2444 L 4447 2448 L 4453 2453 L 4459 2458 L 4465 2462 L 4471 2467 L 4477 2471 L 4483 2476 L 4489 2480 L 4495 2485 L 4501 2489 L 4507 2494 L 4513 2498 L 4519 2503 L 4525 2507 L 4531 2512 L 4537 2516 L 4543 2521 L 4549 2525 L 4555 2530 L 4561 2534 L 4567 2539 L 4573 2544 L 4579 2548 L 4585 2553 L 4591 2557 L 4597 2562 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5097 2941 L 5103 2946 L 5109 2950 L 5115 2955 L 5121 2959 L 5127 2964 L 5133 2968 L 5139 2973 L 5145 2977 L 5151 2982 L 5157 2986 L 5163 2991 L 5169 2995 L 5175 3000 L 5181 3004 L 5187 3009 L 5193 3013 L 5199 3018 L 5205 3022 L 5211 3027 L 5216 3031 L 5222 3036 L 5228 3040 L 5234 3044 L 5240 3049 L 5246 3053 L 5252 3058 L 5258 3062 L 5264 3067 L 5270 3071 L 5276 3076 L 5282 3080 L 5288 3085 L 5294 3089 L 5300 3094 L 5306 3098 L 5312 3103 L 5318 3107 L 5324 3112 L 5330 3116 L 5336 3121 L 5342 3125 L 5348 3130 L 5354 3134 L 5360 3139 L 5365 3143 L 5371 3147 L 5377 3152 L 5383 3156 L 5389 3161 L 5395 3165 L 5401 3170 L 5407 3174 L 5413 3179 L 5419 3183 L 5425 3188 L 5431 3192 L 5437 3197 L 5443 3201 L 5449 3206 L 5455 3210 L 5461 3214 L 5467 3219 L 5473 3223 L 5479 3228 L 5485 3232 L 5491 3237 L 5497 3241 L 5503 3246 L 5509 3250 L 5515 3255 L 5520 3259 L 5526 3263 L 5532 3268 L 5538 3272 L 5544 3277 L 5550 3281 L 5556 3286 L 5562 3290 L 5568 3295 L 5574 3299 L 5580 3303 L 5586 3308 L 5592 3312 L 5598 3317 L 5604 3321 L 5610 3326 L 5616 3330 L 5622 3335 L 5628 3339 L 5634 3343 L 5640 3348 L 5646 3352 L 5652 3357 L 5658 3361 L 5664 3366 L 5670 3370 L 5675 3374 L 5681 3379 L 5687 3383 L 5693 3388 L 5699 3392 L 5705 3397 L 5711 3401 L 5717 3405 L 5723 3410 L 5729 3414 L 5735 3419 L 5741 3423 L 5747 3427 L 5753 3432 L 5759 3436 L currentpoint stroke moveto 5765 3441 L 5771 3445 L 5777 3450 L 5783 3454 L 5789 3458 L 5795 3463 L 5801 3467 L 5807 3472 L 5813 3476 L 5819 3480 L 5824 3485 L 5830 3489 L 5836 3494 L 5842 3498 L 5848 3502 L 5854 3507 L 5860 3511 L 5866 3516 L 5872 3520 L 5878 3524 L 5884 3529 L 5890 3533 L 5896 3537 L 5902 3542 L 5908 3546 L 5914 3551 L 5920 3555 L 5926 3559 L 5932 3564 L 5938 3568 L 5944 3573 L 5950 3577 L 5956 3581 L 5962 3586 L 5968 3590 L 5974 3594 L 5979 3599 L 5985 3603 L 5991 3608 L 5997 3612 L 6003 3616 L 6009 3621 L 6015 3625 L 6021 3629 L 6027 3634 L 6033 3638 L 6039 3643 L 6045 3647 L 6051 3651 L 6057 3656 L 6063 3660 L 6069 3664 L 6075 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L 6576 4030 L 6582 4035 L 6587 4039 L 6593 4043 L 6599 4047 L 6605 4052 L 6611 4056 L 6617 4060 L 6623 4064 L 6629 4068 L 6635 4073 L 6641 4077 L 6647 4081 L 6653 4085 L 6659 4090 L 6665 4094 L 6671 4098 L 6677 4102 L 6683 4107 L 6689 4111 L 6695 4115 L 6701 4119 L 6707 4124 L 6713 4128 L 6719 4132 L 6725 4136 L 6731 4140 L 6737 4145 L 6742 4149 L 6748 4153 L 6754 4157 L 6760 4161 L 6766 4166 L 6772 4170 L 6778 4174 L 6784 4178 L 6790 4183 L 6796 4187 L 6802 4191 L 6808 4195 L 6814 4199 L 6820 4204 L 6826 4208 L 6832 4212 L 6838 4216 L 6844 4220 L 6850 4224 L 6856 4229 L 6862 4233 L 6868 4237 L 6874 4241 L 6880 4245 L 6886 4250 L 6892 4254 L 6897 4258 L 6903 4262 L 6909 4266 L 6915 4271 L 6921 4275 L 6927 4279 L 6933 4283 L 6939 4287 L 6945 4291 L 6951 4296 L 6957 4300 L 6963 4304 L 6969 4308 L LT1 1400 4276 M Th (5/3) show 1100 4346 M 1350 4346 L 1014 491 M 1014 491 L 1020 491 L 1026 491 L 1032 491 L 1038 491 L 1044 491 L 1050 491 L 1056 491 L 1062 491 L 1068 491 L 1074 491 L 1080 491 L 1085 491 L 1091 491 L 1097 491 L 1103 491 L 1109 491 L 1115 491 L 1121 491 L 1127 491 L 1133 491 L 1139 491 L 1145 491 L 1151 491 L 1157 491 L 1163 491 L 1169 491 L 1175 491 L 1181 491 L 1187 491 L 1193 491 L 1199 491 L 1205 492 L 1211 492 L 1217 492 L 1223 492 L 1229 492 L 1235 492 L 1240 492 L 1246 492 L 1252 492 L 1258 492 L 1264 492 L 1270 492 L 1276 492 L 1282 492 L 1288 492 L 1294 492 L 1300 492 L 1306 492 L 1312 492 L 1318 492 L 1324 492 L 1330 492 L 1336 492 L 1342 492 L 1348 493 L 1354 493 L 1360 493 L 1366 493 L 1372 493 L 1378 493 L 1384 493 L 1390 493 L 1395 493 L 1401 493 L 1407 493 L 1413 493 L 1419 493 L 1425 493 L 1431 493 L 1437 493 L 1443 494 L 1449 494 L 1455 494 L 1461 494 L 1467 494 L 1473 494 L 1479 494 L 1485 494 L 1491 494 L 1497 494 L 1503 494 L 1509 494 L 1515 494 L 1521 494 L 1527 495 L 1533 495 L 1539 495 L 1544 495 L 1550 495 L 1556 495 L 1562 495 L 1568 495 L 1574 495 L 1580 495 L 1586 495 L 1592 496 L 1598 496 L 1604 496 L 1610 496 L 1616 496 L 1622 496 L 1628 496 L 1634 496 L 1640 496 L 1646 496 L 1652 496 L 1658 497 L 1664 497 L 1670 497 L 1676 497 L 1682 497 L 1688 497 L 1694 497 L 1699 497 L 1705 497 L 1711 498 L 1717 498 L 1723 498 L 1729 498 L 1735 498 L 1741 498 L 1747 498 L 1753 498 L 1759 498 L 1765 499 L 1771 499 L 1777 499 L 1783 499 L 1789 499 L 1795 499 L 1801 499 L 1807 499 L 1813 499 L 1819 500 L 1825 500 L 1831 500 L 1837 500 L 1843 500 L 1849 500 L 1854 500 L 1860 501 L 1866 501 L 1872 501 L 1878 501 L 1884 501 L 1890 501 L 1896 501 L 1902 501 L 1908 502 L 1914 502 L 1920 502 L 1926 502 L 1932 502 L 1938 502 L 1944 502 L 1950 503 L 1956 503 L 1962 503 L 1968 503 L 1974 503 L 1980 503 L 1986 503 L 1992 504 L 1998 504 L 2003 504 L 2009 504 L 2015 504 L 2021 504 L 2027 505 L 2033 505 L 2039 505 L 2045 505 L 2051 505 L 2057 505 L 2063 506 L 2069 506 L 2075 506 L 2081 506 L 2087 506 L 2093 506 L 2099 507 L 2105 507 L 2111 507 L 2117 507 L 2123 507 L 2129 508 L 2135 508 L 2141 508 L 2147 508 L 2153 509 L 2158 509 L 2164 509 L 2170 509 L 2176 509 L 2182 510 L 2188 510 L 2194 510 L 2200 511 L 2206 511 L 2212 511 L 2218 511 L 2224 512 L 2230 512 L 2236 512 L 2242 513 L 2248 513 L 2254 513 L 2260 514 L 2266 514 L 2272 514 L 2278 515 L 2284 515 L 2290 515 L 2296 516 L 2302 516 L 2307 516 L 2313 517 L 2319 517 L 2325 518 L 2331 518 L 2337 518 L 2343 519 L 2349 519 L 2355 520 L 2361 520 L 2367 521 L 2373 521 L 2379 521 L 2385 522 L 2391 522 L 2397 523 L 2403 523 L 2409 524 L 2415 524 L 2421 525 L 2427 525 L 2433 526 L 2439 526 L 2445 527 L 2451 527 L 2457 528 L 2462 529 L 2468 529 L 2474 530 L 2480 530 L 2486 531 L 2492 531 L 2498 532 L 2504 533 L 2510 533 L 2516 534 L 2522 534 L 2528 535 L 2534 536 L 2540 536 L 2546 537 L 2552 538 L 2558 538 L 2564 539 L 2570 540 L 2576 540 L 2582 541 L 2588 542 L 2594 542 L 2600 543 L 2606 544 L 2612 545 L 2617 545 L 2623 546 L 2629 547 L 2635 547 L 2641 548 L 2647 549 L 2653 550 L 2659 550 L 2665 551 L 2671 552 L 2677 553 L 2683 554 L 2689 554 L 2695 555 L 2701 556 L 2707 557 L 2713 558 L 2719 558 L 2725 559 L 2731 560 L 2737 561 L 2743 562 L 2749 563 L 2755 563 L 2761 564 L 2766 565 L 2772 566 L 2778 567 L 2784 568 L 2790 569 L 2796 570 L 2802 570 L 2808 571 L 2814 572 L 2820 573 L 2826 574 L 2832 575 L 2838 576 L 2844 577 L 2850 578 L 2856 579 L 2862 580 L 2868 581 L 2874 582 L 2880 583 L 2886 584 L 2892 585 L 2898 586 L 2904 587 L 2910 588 L 2916 589 L 2921 590 L 2927 591 L 2933 592 L 2939 593 L 2945 594 L 2951 595 L 2957 596 L 2963 597 L 2969 598 L 2975 599 L 2981 600 L 2987 601 L 2993 602 L 2999 603 L 3005 604 L 3011 605 L 3017 606 L 3023 607 L 3029 608 L 3035 609 L 3041 611 L 3047 612 L 3053 613 L 3059 614 L 3065 615 L 3071 616 L 3076 617 L 3082 618 L 3088 620 L 3094 621 L 3100 622 L 3106 623 L 3112 624 L 3118 625 L 3124 626 L 3130 628 L 3136 629 L 3142 630 L 3148 631 L 3154 632 L 3160 634 L 3166 635 L 3172 636 L 3178 637 L 3184 638 L 3190 640 L 3196 641 L 3202 642 L 3208 643 L 3214 645 L 3220 646 L 3225 647 L 3231 648 L 3237 650 L 3243 651 L 3249 652 L 3255 653 L 3261 655 L 3267 656 L 3273 657 L 3279 658 L 3285 660 L 3291 661 L 3297 662 L 3303 664 L 3309 665 L 3315 666 L 3321 668 L 3327 669 L 3333 670 L 3339 672 L 3345 673 L 3351 674 L 3357 676 L 3363 677 L 3369 678 L 3375 680 L currentpoint stroke moveto 3380 681 L 3386 683 L 3392 684 L 3398 685 L 3404 687 L 3410 688 L 3416 690 L 3422 691 L 3428 693 L 3434 694 L 3440 695 L 3446 697 L 3452 698 L 3458 700 L 3464 701 L 3470 703 L 3476 704 L 3482 706 L 3488 707 L 3494 709 L 3500 710 L 3506 712 L 3512 713 L 3518 715 L 3524 716 L 3530 718 L 3535 719 L 3541 721 L 3547 722 L 3553 724 L 3559 725 L 3565 727 L 3571 728 L 3577 730 L 3583 731 L 3589 733 L 3595 735 L 3601 736 L 3607 738 L 3613 739 L 3619 741 L 3625 743 L 3631 744 L 3637 746 L 3643 747 L 3649 749 L 3655 751 L 3661 752 L 3667 754 L 3673 756 L 3679 757 L 3684 759 L 3690 761 L 3696 762 L 3702 764 L 3708 766 L 3714 767 L 3720 769 L 3726 771 L 3732 772 L 3738 774 L 3744 776 L 3750 777 L 3756 779 L 3762 781 L 3768 783 L 3774 784 L 3780 786 L 3786 788 L 3792 790 L 3798 791 L 3804 793 L 3810 795 L 3816 797 L 3822 798 L 3828 800 L 3834 802 L 3839 804 L 3845 805 L 3851 807 L 3857 809 L 3863 811 L 3869 813 L 3875 814 L 3881 816 L 3887 818 L 3893 820 L 3899 822 L 3905 824 L 3911 825 L 3917 827 L 3923 829 L 3929 831 L 3935 833 L 3941 835 L 3947 837 L 3953 838 L 3959 840 L 3965 842 L 3971 844 L 3977 846 L 3983 848 L 3989 850 L 3994 852 L 4000 854 L 4006 855 L 4012 857 L 4018 859 L 4024 861 L 4030 863 L 4036 865 L 4042 867 L 4048 869 L 4054 871 L 4060 873 L 4066 875 L 4072 877 L 4078 879 L 4084 881 L 4090 883 L 4096 885 L 4102 887 L 4108 889 L 4114 891 L 4120 893 L 4126 895 L 4132 897 L 4138 899 L 4143 901 L 4149 903 L 4155 905 L 4161 907 L 4167 909 L 4173 911 L 4179 913 L 4185 915 L 4191 917 L 4197 919 L 4203 921 L 4209 923 L 4215 925 L 4221 927 L 4227 929 L 4233 931 L 4239 933 L 4245 935 L 4251 937 L 4257 940 L 4263 942 L 4269 944 L 4275 946 L 4281 948 L 4287 950 L 4293 952 L 4298 954 L 4304 956 L 4310 958 L 4316 961 L 4322 963 L 4328 965 L 4334 967 L 4340 969 L 4346 971 L 4352 973 L 4358 975 L 4364 978 L 4370 980 L 4376 982 L 4382 984 L 4388 986 L 4394 988 L 4400 991 L 4406 993 L 4412 995 L 4418 997 L 4424 999 L 4430 1001 L 4436 1004 L 4442 1006 L 4447 1008 L 4453 1010 L 4459 1012 L 4465 1015 L 4471 1017 L 4477 1019 L 4483 1021 L 4489 1023 L 4495 1026 L 4501 1028 L 4507 1030 L 4513 1032 L 4519 1035 L 4525 1037 L 4531 1039 L 4537 1041 L 4543 1044 L 4549 1046 L 4555 1048 L 4561 1050 L 4567 1053 L 4573 1055 L 4579 1057 L 4585 1059 L 4591 1062 L 4597 1064 L 4602 1066 L 4608 1068 L 4614 1071 L 4620 1073 L 4626 1075 L 4632 1077 L 4638 1080 L 4644 1082 L 4650 1084 L 4656 1087 L 4662 1089 L 4668 1091 L 4674 1094 L 4680 1096 L 4686 1098 L 4692 1100 L 4698 1103 L 4704 1105 L 4710 1107 L 4716 1110 L 4722 1112 L 4728 1114 L 4734 1117 L 4740 1119 L 4746 1121 L 4752 1124 L 4757 1126 L 4763 1128 L 4769 1131 L 4775 1133 L 4781 1135 L 4787 1138 L 4793 1140 L 4799 1142 L 4805 1145 L 4811 1147 L 4817 1150 L 4823 1152 L 4829 1154 L 4835 1157 L 4841 1159 L 4847 1161 L 4853 1164 L 4859 1166 L 4865 1168 L 4871 1171 L 4877 1173 L 4883 1176 L 4889 1178 L 4895 1180 L 4901 1183 L 4906 1185 L 4912 1188 L 4918 1190 L 4924 1192 L 4930 1195 L 4936 1197 L 4942 1200 L 4948 1202 L 4954 1204 L 4960 1207 L 4966 1209 L 4972 1212 L 4978 1214 L 4984 1216 L 4990 1219 L 4996 1221 L 5002 1224 L 5008 1226 L 5014 1229 L 5020 1231 L 5026 1233 L 5032 1236 L 5038 1238 L 5044 1241 L 5050 1243 L 5056 1246 L 5061 1248 L 5067 1251 L 5073 1253 L 5079 1255 L 5085 1258 L 5091 1260 L 5097 1263 L 5103 1265 L 5109 1268 L 5115 1270 L 5121 1273 L 5127 1275 L 5133 1278 L 5139 1280 L 5145 1283 L 5151 1285 L 5157 1287 L 5163 1290 L 5169 1292 L 5175 1295 L 5181 1297 L 5187 1300 L 5193 1302 L 5199 1305 L 5205 1307 L 5211 1310 L 5216 1312 L 5222 1315 L 5228 1317 L 5234 1320 L 5240 1322 L 5246 1325 L 5252 1327 L 5258 1330 L 5264 1332 L 5270 1335 L 5276 1337 L 5282 1340 L 5288 1342 L 5294 1345 L 5300 1347 L 5306 1350 L 5312 1352 L 5318 1355 L 5324 1357 L 5330 1360 L 5336 1362 L 5342 1365 L 5348 1367 L 5354 1370 L 5360 1372 L 5365 1375 L 5371 1377 L 5377 1380 L 5383 1382 L 5389 1385 L 5395 1387 L 5401 1390 L 5407 1393 L 5413 1395 L 5419 1398 L 5425 1400 L 5431 1403 L 5437 1405 L 5443 1408 L 5449 1410 L 5455 1413 L 5461 1415 L 5467 1418 L 5473 1420 L 5479 1423 L 5485 1426 L 5491 1428 L 5497 1431 L 5503 1433 L 5509 1436 L 5515 1438 L 5520 1441 L 5526 1443 L 5532 1446 L 5538 1448 L 5544 1451 L 5550 1454 L 5556 1456 L 5562 1459 L 5568 1461 L 5574 1464 L 5580 1466 L 5586 1469 L 5592 1472 L 5598 1474 L 5604 1477 L 5610 1479 L 5616 1482 L 5622 1484 L 5628 1487 L 5634 1490 L 5640 1492 L 5646 1495 L 5652 1497 L 5658 1500 L 5664 1502 L 5670 1505 L 5675 1508 L 5681 1510 L 5687 1513 L 5693 1515 L 5699 1518 L 5705 1520 L 5711 1523 L 5717 1526 L 5723 1528 L 5729 1531 L 5735 1533 L 5741 1536 L 5747 1539 L 5753 1541 L 5759 1544 L currentpoint stroke moveto 5765 1546 L 5771 1549 L 5777 1551 L 5783 1554 L 5789 1557 L 5795 1559 L 5801 1562 L 5807 1564 L 5813 1567 L 5819 1570 L 5824 1572 L 5830 1575 L 5836 1577 L 5842 1580 L 5848 1583 L 5854 1585 L 5860 1588 L 5866 1590 L 5872 1593 L 5878 1596 L 5884 1598 L 5890 1601 L 5896 1603 L 5902 1606 L 5908 1609 L 5914 1611 L 5920 1614 L 5926 1617 L 5932 1619 L 5938 1622 L 5944 1624 L 5950 1627 L 5956 1630 L 5962 1632 L 5968 1635 L 5974 1637 L 5979 1640 L 5985 1643 L 5991 1645 L 5997 1648 L 6003 1650 L 6009 1653 L 6015 1656 L 6021 1658 L 6027 1661 L 6033 1664 L 6039 1666 L 6045 1669 L 6051 1671 L 6057 1674 L 6063 1677 L 6069 1679 L 6075 1682 L 6081 1685 L 6087 1687 L 6093 1690 L 6099 1692 L 6105 1695 L 6111 1698 L 6117 1700 L 6123 1703 L 6128 1706 L 6134 1708 L 6140 1711 L 6146 1713 L 6152 1716 L 6158 1719 L 6164 1721 L 6170 1724 L 6176 1727 L 6182 1729 L 6188 1732 L 6194 1734 L 6200 1737 L 6206 1740 L 6212 1742 L 6218 1745 L 6224 1748 L 6230 1750 L 6236 1753 L 6242 1755 L 6248 1758 L 6254 1761 L 6260 1763 L 6266 1766 L 6272 1769 L 6278 1771 L 6283 1774 L 6289 1777 L 6295 1779 L 6301 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L 2415 500 L 2421 501 L 2427 501 L 2433 502 L 2439 502 L 2445 503 L 2451 503 L 2457 504 L 2462 504 L 2468 505 L 2474 505 L 2480 506 L 2486 506 L 2492 507 L 2498 507 L 2504 508 L 2510 508 L 2516 509 L 2522 509 L 2528 510 L 2534 511 L 2540 511 L 2546 512 L 2552 512 L 2558 513 L 2564 514 L 2570 514 L 2576 515 L 2582 516 L 2588 516 L 2594 517 L 2600 518 L 2606 518 L 2612 519 L 2617 520 L 2623 520 L 2629 521 L 2635 522 L 2641 523 L 2647 523 L 2653 524 L 2659 525 L 2665 526 L 2671 526 L 2677 527 L 2683 528 L 2689 529 L 2695 529 L 2701 530 L 2707 531 L 2713 532 L 2719 532 L 2725 533 L 2731 534 L 2737 535 L 2743 536 L 2749 537 L 2755 538 L 2761 539 L 2766 539 L 2772 540 L 2778 541 L 2784 542 L 2790 543 L 2796 544 L 2802 545 L 2808 546 L 2814 547 L 2820 548 L 2826 549 L 2832 550 L 2838 550 L 2844 551 L 2850 552 L 2856 553 L 2862 554 L 2868 555 L 2874 556 L 2880 557 L 2886 558 L 2892 559 L 2898 560 L 2904 561 L 2910 563 L 2916 564 L 2921 565 L 2927 566 L 2933 567 L 2939 568 L 2945 569 L 2951 570 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5538 1596 L 5544 1599 L 5550 1602 L 5556 1605 L 5562 1609 L 5568 1612 L 5574 1615 L 5580 1618 L 5586 1622 L 5592 1625 L 5598 1628 L 5604 1631 L 5610 1635 L 5616 1638 L 5622 1641 L 5628 1644 L 5634 1648 L 5640 1651 L 5646 1654 L 5652 1658 L 5658 1661 L 5664 1664 L 5670 1667 L 5675 1671 L 5681 1674 L 5687 1677 L 5693 1681 L 5699 1684 L 5705 1687 L 5711 1690 L 5717 1694 L 5723 1697 L 5729 1700 L 5735 1704 L 5741 1707 L 5747 1710 L 5753 1714 L 5759 1717 L currentpoint stroke moveto 5765 1720 L 5771 1723 L 5777 1727 L 5783 1730 L 5789 1733 L 5795 1737 L 5801 1740 L 5807 1743 L 5813 1747 L 5819 1750 L 5824 1753 L 5830 1757 L 5836 1760 L 5842 1763 L 5848 1767 L 5854 1770 L 5860 1773 L 5866 1777 L 5872 1780 L 5878 1783 L 5884 1787 L 5890 1790 L 5896 1790 L 5902 1793 L 5908 1797 L 5914 1800 L 5920 1803 L 5926 1807 L 5932 1810 L 5938 1813 L 5944 1817 L 5950 1820 L 5956 1823 L 5962 1827 L 5968 1830 L 5974 1834 L 5979 1837 L 5985 1840 L 5991 1844 L 5997 1847 L 6003 1850 L 6009 1854 L 6015 1857 L 6021 1860 L 6027 1864 L 6033 1867 L 6039 1870 L 6045 1874 L 6051 1877 L 6057 1880 L 6063 1884 L 6069 1887 L 6075 1891 L 6081 1894 L 6087 1897 L 6093 1901 L 6099 1904 L 6105 1907 L 6111 1911 L 6117 1914 L 6123 1918 L 6128 1921 L 6134 1924 L 6140 1928 L 6146 1931 L 6152 1931 L 6158 1934 L 6164 1938 L 6170 1941 L 6176 1944 L 6182 1948 L 6188 1951 L 6194 1955 L 6200 1958 L 6206 1961 L 6212 1965 L 6218 1968 L 6224 1971 L 6230 1975 L 6236 1978 L 6242 1982 L 6248 1985 L 6254 1988 L 6260 1988 L 6266 1992 L 6272 1995 L 6278 1998 L 6283 2002 L 6289 2005 L 6295 2008 L 6301 2012 L 6307 2015 L 6313 2019 L 6319 2022 L 6325 2025 L 6331 2029 L 6337 2032 L 6343 2035 L 6349 2039 L 6355 2042 L 6361 2046 L 6367 2049 L 6373 2052 L 6379 2056 L 6385 2059 L 6391 2063 L 6397 2066 L 6403 2066 L 6409 2069 L 6415 2073 L 6421 2076 L 6427 2079 L 6433 2083 L 6438 2086 L 6444 2089 L 6450 2093 L 6456 2096 L 6462 2100 L 6468 2103 L 6474 2106 L 6480 2110 L 6486 2113 L 6492 2117 L 6498 2120 L 6504 2123 L 6510 2127 L 6516 2130 L 6522 2130 L 6528 2133 L 6534 2137 L 6540 2140 L 6546 2143 L 6552 2147 L 6558 2150 L 6564 2154 L 6570 2157 L 6576 2160 L 6582 2164 L 6587 2167 L 6593 2170 L 6599 2174 L 6605 2177 L 6611 2181 L 6617 2184 L 6623 2187 L 6629 2191 L 6635 2194 L 6641 2198 L 6647 2201 L 6653 2204 L 6659 2208 L 6665 2211 L 6671 2215 L 6677 2218 L 6683 2221 L 6689 2225 L 6695 2228 L 6701 2232 L 6707 2235 L 6713 2238 L 6719 2242 L 6725 2245 L 6731 2249 L 6737 2252 L 6742 2255 L 6748 2259 L 6754 2262 L 6760 2266 L 6766 2269 L 6772 2272 L 6778 2276 L 6784 2279 L 6790 2283 L 6796 2286 L 6802 2289 L 6808 2289 L 6814 2292 L 6820 2296 L 6826 2299 L 6832 2302 L 6838 2306 L 6844 2309 L 6850 2313 L 6856 2316 L 6862 2319 L 6868 2323 L 6874 2326 L 6880 2329 L 6886 2333 L 6892 2336 L 6897 2340 L 6903 2343 L 6909 2346 L 6915 2350 L 6921 2353 L 6927 2357 L 6933 2360 L 6939 2363 L 6945 2367 L 6951 2370 L 6957 2374 L 6963 2377 L 6969 2380 L LT3 1400 3996 M Th (1.618033989) show 1100 4066 M 1350 4066 L 1014 491 M 1014 491 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L 1562 491 L 1568 491 L 1574 491 L 1580 491 L 1586 491 L 1592 491 L 1598 491 L 1604 491 L 1610 491 L 1616 491 L 1622 491 L 1628 491 L 1634 491 L 1640 491 L 1646 491 L 1652 491 L 1658 491 L 1664 491 L 1670 491 L 1676 491 L 1682 491 L 1688 491 L 1694 491 L 1699 491 L 1705 491 L 1711 491 L 1717 491 L 1723 491 L 1729 491 L 1735 491 L 1741 491 L 1747 491 L 1753 491 L 1759 491 L 1765 491 L 1771 491 L 1777 491 L 1783 491 L 1789 491 L 1795 491 L 1801 491 L 1807 491 L 1813 491 L 1819 491 L 1825 491 L 1831 491 L 1837 491 L 1843 491 L 1849 491 L 1854 491 L 1860 491 L 1866 491 L 1872 491 L 1878 491 L 1884 491 L 1890 491 L 1896 491 L 1902 491 L 1908 491 L 1914 491 L 1920 491 L 1926 491 L 1932 491 L 1938 491 L 1944 491 L 1950 491 L 1956 491 L 1962 491 L 1968 491 L 1974 491 L 1980 491 L 1986 491 L 1992 491 L 1998 491 L 2003 491 L 2009 491 L 2015 491 L 2021 491 L 2027 491 L 2033 491 L 2039 491 L 2045 491 L 2051 491 L 2057 491 L 2063 491 L 2069 491 L 2075 491 L 2081 491 L 2087 491 L 2093 491 L 2099 491 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 241 x Fw(Figure)27 b(9:)41 b(The)28 b(coupling)g(constan)m (t)g(plot)f(for)g(the)44 b Fn(\016)2117 5176 y Fq(0)2113 5237 y Fv(s)2193 5212 y Fw(lattice)28 b(sp)s(ectrum)h(in)e(the)h (golden)f(mean)165 5332 y(case.)1888 5682 y(29)p eop %%Page: 30 30 30 29 bop 165 4665 a @beginspecial 9 @llx 60 @lly 473 @urx 508 @ury 4640 @rwi @setspecial %%BeginDocument: Bild10_2.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 241 x Fw(Figure)33 b(10:)43 b(The)50 b Fn(\016)905 4870 y Fq(0)901 4930 y Fv(s)987 4906 y Fw(lattice)33 b(sp)s(ectrum)h(as)f(a)f(function)h(of)f(the)50 b Fn(\022)i Fw(for)d Fn(\014)33 b Fw(=)28 b(5)17 b(.)1888 5682 y(30)p eop %%Page: 31 31 31 30 bop 165 4569 a @beginspecial -5 @llx 93 @lly 453 @urx 531 @ury 4580 @rwi @setspecial %%BeginDocument: Bild11.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def /rgbindx 0 def -7 25 translate 612.00000 792.00000 scale 612 792 1 [612 0 0 -792 0 792] {currentfile pix readhexstring pop} image ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 240 x Fw(Figure)34 b(11:)45 b(The)34 b(folded)50 b Fn(\016)1199 4773 y Fq(0)1195 4834 y Fv(s)1282 4809 y Fw(lattice)34 b(sp)s(ectrum)h(as)f(a)f(function)h(of)f(the)51 b Fn(\022)i Fw(for)c Fn(\013)30 b Fw(=)f(20)17 b(.)46 b(The)165 4930 y(v)-5 b(alue)33 b(of)f(momen)m(tum)j(is)e(restricted)h (to)49 b Fn(k)30 b Fm(\024)f Fw(100)17 b(.)1888 5682 y(31)p eop %%Page: 32 32 32 31 bop 165 4688 a @beginspecial -1 @llx 68 @lly 460 @urx 521 @ury 4610 @rwi @setspecial %%BeginDocument: Bild12_1.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def /rgbindx 0 def 0 0 translate 612.00000 792.00000 scale 612 792 1 [612 0 0 -792 0 792] {currentfile pix readhexstring pop} image ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff 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Fw(for)g(rational)g(appro)m(ximations)i(to)f(the)g(golden)g (mean:)165 4929 y Fn(\014)i Fw(=)27 b(20)p Fn(;)17 b Fw(.)1888 5682 y(32)p eop %%Page: 33 33 33 32 bop 165 4628 a @beginspecial -1 @llx 68 @lly 460 @urx 521 @ury 4610 @rwi @setspecial %%BeginDocument: Bild12_2.epsi /InitDictCount countdictstack def gsave save mark newpath /showpage {} def /origstate save def 20 dict begin /pix 77 string def /grays 612 string def % space for gray scale line /npixls 0 def /rgbindx 0 def 0 0 translate 612.00000 792.00000 scale 612 792 1 [612 0 0 -792 0 792] {currentfile pix readhexstring pop} image ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffff end origstate restore countdictstack InitDictCount sub { end } repeat cleartomark restore grestore %%EndDocument @endspecial 241 x Fw(Figure)32 b(12b:)42 b(F)-8 b(olded{sp)s(ectrum)33 b Fo(gaps)d Fw(for)h(rational)f(appro)m(ximations)j(to)e(the)g(golden)h (mean:)165 4989 y Fn(\014)i Fw(=)27 b(5)17 b(.)1888 5682 y(33)p eop %%Page: 34 34 34 33 bop 331 218 a Fw(One)38 b(lesson)i(dra)m(wn)e(from)g(the)h (presen)m(t)h(considerations)f(concerns)h(the)e(v)-5 b(alidit)m(y)40 b(of)d(the)165 338 y(so{called)46 b Fo (Bethe{Sommerfeld)e(c)-5 b(onje)g(ctur)g(e)45 b Fw(b)m(y)g(whic)m(h)i (the)e(n)m(um)m(b)s(er)i(of)d(gaps)h(in)g(p)s(erio)s(dic)165 459 y(systems)e(of)d(dimension)i(t)m(w)m(o)f(or)f(higher)g(is)h (\014nite.)68 b(This)41 b(w)m(as)g(pro)m(v)m(en)h(b)m(y)f(Skrigano)m(v) 3476 422 y Fr(30)3592 459 y Fw(for)165 579 y(Sc)m(hr\177)-49 b(odinger)30 b(op)s(erators)e(with)h(nice)g(p)s(oten)m(tials,)h(and)e (it)g(is)h(kno)m(wn)g(to)f(b)s(e)h(v)-5 b(alid)28 b(also)g(for)g(some) 165 699 y(systems)35 b(with)48 b Fn(\016)53 b Fw(in)m(teractions.)1382 663 y Fr(2)1454 699 y Fw(W)-8 b(e)32 b(ha)m(v)m(e)h(seen)g(that)f(the)g (conjecture)h(ma)m(y)f(not)g(b)s(e)g(v)-5 b(alid)32 b(if)165 820 y(the)c(p)s(erio)s(dic)f(structure)h(is)f(singular)h(enough:)41 b(the)27 b(golden{mean)44 b Fn(\016)k Fw(lattice)27 b(has,)h(dep)s (ending)165 940 y(on)k(the)g(v)-5 b(alue)32 b(of)f(the)h(coupling)g (constan)m(t)49 b Fn(\013)18 b Fw(,)32 b(either)g(in\014nitely)h(man)m (y)g(gaps)f(or)f Fo(none)i(at)h(al)5 b(l.)195 1179 y Fw(The)30 b(w)m(ork)h(w)m(as)f(done)g(during)g(the)g(visits)h(of)e(P)-8 b(.E.)30 b(at)g(the)g(Institut)g(f)s(\177)-51 b(ur)28 b(Mathematik,)k(Ruhr{)165 1299 y(Univ)m(ersit\177)-49 b(at)36 b(Bo)s(c)m(h)m(um)g(and)e(R.G.)g(at)g(the)h(Nuclear)g(Ph)m (ysics)i(Institute,)f(Czec)m(h)g(Academ)m(y)g(of)165 1419 y(Sciences;)45 b(the)40 b(authors)f(express)i(their)f(gratitude)f (to)g(the)h(hosts.)64 b(The)40 b(researc)m(h)g(has)g(b)s(een)165 1540 y(partially)d(supp)s(orted)g(b)m(y)h(the)f(the)g(Gran)m(ts)f(AS)h (No.148409)e(and)h(GA)m(CR)h(No.202{93{1314,)1888 5682 y(34)p eop %%Page: 35 35 35 34 bop 165 218 a Fw(and)33 b(the)g(Europ)s(ean)g(Union)g(Pro)5 b(ject)33 b(ERB{CIP)-8 b(A{3510{CT{920704/704.)p 165 852 1418 4 v 278 913 a Fe(1)315 943 y Fd(The)23 b(literature)e(ab)r (out)i(this)g(sub)5 b(ject)23 b(is)f(h)n(uge;)i(let)f(us)g(just)g(men)n (tion)g(the)g(classical)e(pap)r(ers:)34 b(P)-7 b(.G.)22 b(Harp)r(er,)165 1043 y(Pro)r(c.Ro)n(y)-7 b(.So)r(c.\(London\))29 b(A)j Fc(68)p Fd(,)f(874)f(\(1955\);)i(D.R.)f(Hofstadter,)h(Ph)n (ys.Rev.)46 b(B)31 b Fc(14)p Fd(,)h(2239)d(\(1976\).)46 b(F)-7 b(or)30 b(a)165 1142 y(more)f(complete)g(bibilograph)n(y)f(to)h (this)h(problem,)f(whic)n(h)g(is)h(usually)e(referred)h(to)g(as)g(the)g (\\almost)g(Mathieu)165 1242 y(equation",)d(see)f(Ref.2,)i(Sec.I)r(I)r (I.2.5)f(or)f(Ref.3;)h(recen)n(t)g(mathematical)f(results)h(are)f (discussed,)h Fb(e.g.)p Fd(,)i(in)e(Refs.4)165 1341 y(and)i(5.)278 1411 y Fe(2)315 1441 y Fd(S.)d(Alb)r(ev)n(erio,)f(F.)h(Gesztesy)-7 b(,)25 b(R.)g(H\034egh{Krohn,)e(H.)i(Holden:)35 b Fb(Solvable)29 b(Mo)l(dels)f(in)g(Quantum)d(Me)l(chan-)165 1541 y(ics,)k Fd(Springer{V)-7 b(erlag,)25 b(Heidelb)r(erg)i(1988.)278 1610 y Fe(3)315 1640 y Fd(J.)e(Bellissard,)f(in)h Fb(Numb)l(er)i(The)l (ory)h(and)g(Physics,)g Fd(\(M.)51 b(W)-7 b(aldsc)n(hmidt)25 b(et)g(al.,)g(eds.\),)h(Springer,)f(Heidel-)165 1740 y(b)r(er)j(1992;)e(pp.)37 b(538{630.)278 1809 y Fe(4)315 1840 y Fd(M.A.)28 b(Sh)n(ubin,)g(Comm)n(un.Math.Ph)n(ys.)36 b Fc(164)p Fd(,)27 b(259)f(\(1994\).)278 1909 y Fe(5)315 1939 y Fd(Y.)i(Last,)f(Comm)n(un.Math.Ph)n(ys.)36 b Fc(164)p Fd(,)27 b(421)f(\(1994\).)278 2012 y Fe(6)315 2042 y Fd(P)-7 b(.)27 b(Duclos,)h(P)-7 b(.)27 b(Exner,)g(P)-7 b(.)1179 2021 y(\024)1177 2042 y(S)1218 2027 y(\024)1223 2042 y(to)n(v)e(\023)-32 b(\020)n(\024)-39 b(cek,)25 b(Ann.Inst.H.P)n(oincar)n(\023)-39 b(e:)34 b(Ph)n(ys.Th)n(\023)-39 b(eor.)34 b Fc(62)p Fd(,)28 b(81)e(\(1995\))278 2112 y Fe(7)315 2142 y Fd(K.)h(Rueden)n(b)r(erg,)g(C.W.)h(Sc)n(herr,)f (J.Chem.Ph)n(ys.)36 b Fc(21)p Fd(,)27 b(1565)e(\(1953\).)278 2211 y Fe(8)315 2242 y Fd(V.M.)j(Adam)n(y)n(an,)f(Op)r(er.Theory:)35 b(Adv.)i(Appl.)h Fc(59)p Fd(,)27 b(1)g(\(1992\).)278 2311 y Fe(9)315 2341 y Fd(Y.)h(Avishai,)f(J.M.)h(Luc)n(k,)f(Ph)n (ys.Rev.)36 b(B)27 b Fc(45)p Fd(,)g(1074)f(\(1992\).)244 2411 y Fe(10)315 2441 y Fd(J.E.)h(Avron,)g(P)-7 b(.)27 b(Exner,)g(Y.)h(Last,)f(Ph)n(ys.Rev.Lett.)36 b Fc(72)p Fd(,)27 b(896)f(\(1994\).)244 2510 y Fe(11)315 2540 y Fd(J.E.)h(Avron,)g(A.)h(Ra)n(v)n(eh,)e(B.)i(Zur,)f(Rev.Mo)r(d.Ph)n(ys.) 36 b Fc(60)p Fd(,)27 b(873)f(\(1988\).)244 2610 y Fe(12)315 2640 y Fd(W.)i(Bulla,)f(T.)h(T)-7 b(renc)n(kler,)26 b(J.Math.Ph)n(ys.) 36 b Fc(31)p Fd(,)27 b(1157)f(\(1990\).)244 2710 y Fe(13)315 2740 y Fd(P)-7 b(.)27 b(Exner,)g(Ph)n(ys.Rev.Lett.)36 b Fc(74)p Fd(,)27 b(3503)f(\(1995\);)g(J.Ph)n(ys.)36 b(A,)28 b Fb(submitte)l(d)h(for)i(public)l(ation)244 2813 y Fe(14)315 2843 y Fd(P)-7 b(.)27 b(Exner,)g(P)-7 b(.)784 2822 y(\024)782 2843 y(Seba,)27 b(Rep.Math.Ph)n(ys.)37 b Fc(28)p Fd(,)27 b(7)g(\(1989\).)244 2912 y Fe(15)315 2942 y Fd(J.)g(Gratus,)g(C.J.)h(Lam)n(b)r(ert,)f(S.J.)g(Robinson,)g (R.W.)i(T)-7 b(uc)n(k)n(er,)26 b(J.Ph)n(ys.)36 b(A)28 b Fc(27)p Fd(,)f(6881)f(\(1994\).)244 3012 y Fe(16)315 3042 y Fd(N.I.)i(Gerasimenk)n(o,)e(B.S.)h(P)n(a)n(vlo)n(v,)e(So)n (v.J.Theor.Math.Ph)n(ys.)34 b Fc(74)p Fd(,)28 b(345)e(\(1988\).)244 3111 y Fe(17)315 3142 y Fd(A)36 b(single{cen)n(ter)49 b Fh(\016)968 3111 y Fa(0)1027 3142 y Fd(in)n(teraction)36 b(is)g(de\014ned)h(b)n(y)e(the)i(requiremen)n(t)e(of)i(the)f(w)n(a)n(v) n(efunction{deriv)-5 b(ativ)n(e)165 3241 y(con)n(tin)n(uit)n(y)26 b(together)f(with)i(the)f(condition)40 b Fh( )s Fd(\(0+\))15 b Fi(\000)g Fh( )s Fd(\(0)p Fi(\000)p Fd(\))23 b(=)g Fh(\014)t( )2358 3211 y Fa(0)2381 3241 y Fd(\(0\))40 b(for)26 b(some)g(\\coupling)f(constan)n(t")38 b Fh(\014)19 b Fd(.)165 3341 y(A)28 b(thorough)f(discussion)f(of)i(its)g(prop)r (erties)e(can)i(b)r(e)g(found)g(in)f(Ref.2.)244 3410 y Fe(18)315 3440 y Fd(Considerations)38 b(of)h(the)h(presen)n(t)f(pap)r (er)g(allo)n(w)f(a)h(straigh)n(tforw)n(ard)e(extension)i(to)g(graph)f (lattices)h(in)165 3540 y(dimension)j Fh(d)23 b Fi(\025)g Fd(3)14 b(.)244 3610 y Fe(19)315 3640 y Fd(The)36 b(condition)g(\(3\))g (w)n(as)f(obtained)h(in)g(Ref.6)g(for)49 b Fh(\013)38 b Fd(=)e(0)50 b(and)36 b(extended)g(to)g(the)g(nonzero{coupling)165 3739 y(case)24 b(in)h(Ref.12.)36 b(These)24 b(authors)g(pursued)h (other)f(goals,)g(ho)n(w)n(ev)n(er,)f(and)i(therefore)f(restricted)g (themselv)n(es)g(to)165 3839 y(writing)k(do)n(wn)f(the)h(solution)f(in) h(the)g(trivial)f(situation.)244 3908 y Fe(20)315 3939 y Fd(G.H.)41 b(Hardy)-7 b(,)44 b(E.M.)d(W)-7 b(righ)n(t,)44 b Fb(A)n(n)e(Intr)l(o)l(duction)f(to)i(the)f(The)l(ory)h(of)g(Numb)l (ers,)i Fd(Oxford)40 b(Univ)n(ersit)n(y)165 4038 y(Press)27 b(1979;)e(W.M.)k(Sc)n(hmidt,)f Fb(Diophantine)j(Appr)l(oximations)g (and)g(Diophantine)g(Equations,)e Fd(LNM)f(1467,)165 4138 y(Springer{V)-7 b(erlag,)25 b(Berlin)i(1991.)244 4207 y Fe(21)315 4237 y Fd(P)-7 b(.)27 b(Exner,)g Fb(submitte)l(d)i (for)i(public)l(ation)244 4307 y Fe(22)315 4337 y Fd(W)-7 b(e)31 b(use)f(this)h(term)g(in)g(a)f(somewhat)g(lo)r(ose)f(sense)i (without)g(insisting)f(that)h(the)g(folded{gap)e(pattern)i(is)165 4437 y(exactly)c(self{similar.)244 4506 y Fe(23)315 4536 y Fd(M.)h(Maioli,)f(A.)h(Sacc)n(hetti,)f(J.Ph)n(ys.)36 b(A)28 b Fc(28)p Fd(,)f(1101)f(\(1995\))244 4606 y Fe(24)315 4636 y Fd(P)-7 b(.)27 b(Exner,)g(J.Math.Ph)n(ys.)35 b Fc(36)p Fd(,)28 b Fb(to)h(app)l(e)l(ar)244 4705 y Fe(25)315 4736 y Fd(F.)f(Ben)n(tosela,)e(V.)i(Grecc)n(hi,)f(Comm)n(un.Math.Ph)n (ys.)36 b Fc(142)p Fd(,)27 b(169)f(\(1991\),)g(and)i(references)e (therein.)244 4805 y Fe(26)315 4835 y Fd(A.M.)i(Berezhk)n(o)n(vski,)d (A.A.)j(Ov)n(c)n(hinnik)n(o)n(v,)e(So)n(v.Ph)n(ys.:)35 b(Solid)27 b(State)h Fc(18)p Fd(,)f(1908)f(\(1976\))244 4905 y Fe(27)315 4935 y Fd(P)-7 b(.)27 b(Ao,)h(Ph)n(ys.Rev.)35 b(B)28 b Fc(41)p Fd(,)f(3998)f(\(1990\))244 5004 y Fe(28)315 5035 y Fd(T)-7 b(o)21 b(get)f(the)i(stated)f(result)g(from)g(this)g (argumen)n(t,)h(a)e(n)n(umerical)h(factor)f(decribing)g(the)i (tunneling)f(through)165 5134 y(a)28 b(single)f(gap)f(m)n(ust)i(b)r(e)g (prop)r(erly)f(ev)-5 b(aluated)27 b(as)g(p)r(oin)n(ted)h(out)f(b)n(y)h (F.)g(Ben)n(tosela.)244 5204 y Fe(29)315 5234 y Fd(I.P)-7 b(.)27 b(Cornfeld,)g(S.V.)i(F)-7 b(omin,)28 b(Y)-7 b(a.G.)27 b(Sinai:)37 b Fb(Er)l(go)l(dic)31 b(The)l(ory)p Fd(,)f(Springer,)c(New) i(Y)-7 b(ork)27 b(1982;)f(Thm.7.2.1)244 5303 y Fe(30)315 5333 y Fd(M.M.)33 b(Skrigano)n(v,)e(So)n(v.Math.Doklady)g Fc(20)p Fd(,)i(956)e(\(1979\);)j(M.M.)f(Skrigano)n(v,)e(In)n(v)n(en)n (t.Math.)51 b Fc(80)p Fd(,)33 b(107)165 5433 y(\(1985\);)27 b(see)g(also)f(Y)-7 b(u.E.)28 b(Karp)r(eshina,)e(Pro)r(c.)36 b(Steklo)n(v)27 b(Math.Inst.,)h(Issue)f(3,)g(109)g(\(1991\).)1888 5682 y Fw(35)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY