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Bazão, Vanderléa Rodrigues.
B348a Argumentos de Gordon no estudo espectral de operadores de Schrödinger unidimensionais / Vanderléa Rodrigues Bazão. - Presidente Prudente : [s.n], 2012
00 f. Orientador: Roberto de Almeida Prado
Coorientador: Suetônio de Almeida Meira Dissertação (mestrado) - Universidade Estadual Paulista, Faculdade de
Ciências e Tecnologia Inclui bibliografia 1. Operadores de Schrödinger. 2. Teoria Espectral de Operadores. 3.
Argumentos de Gordon. 4. Matemática. I. Prado, Roberto de Almeida. II. Meira, Suetônio de Almeida. III. Universidade Estadual Paulista. Faculdade de Ciências e Tecnologia. IV. Título.
Ficha Catalográfica elaborada pela Seção Técnica de Aquisição e Tratamento da Informação – Serviço Técnico de Biblioteca e Documentação - UNESP, Câmpus de
Presidente Prudente.
Aos meus pais Milton e Janete, dedico!
Agradecimentos
Primeiramente, agradeço a DEUS por iluminar minha vida, guiar meus passos em minhas esco-
lhas e me abençoar com saúde e sabedoria para realizar este trabalho.
Aos meus pais Milton e Janete que são meu alicerce, pelo amor e dedicação em todos os mo-
mentos, sem os quais este sonho não seria possível.
Ao meu amor Gilson pelo carinho, compreensão e por estar sempre ao meu lado em pensamentos,
sonhos e ideais.
Ao Prof. Dr. Roberto de Almeida Prado pela orientação, dedicação e paciência que sempre
teve comigo, pelo apoio e incentivo no desenvolvimento deste trabalho e na continuação de minha
carreira acadêmica.
Ao Prof. Dr. Suetônio de Almeida Meira que além de co-orientar este trabalho me orientou
desde os anos iniciais da graduação, agradeço pela conança que teve em meu trabalho, pelos valiosos
ensinamentos transmitidos, companherismo e amizade, pois sem esse apoio e incentivo tenho certeza
que meu caminhar acadêmico seria menos fecundo.
Aos professores do Programa de Pós-Graduação em Matemática Aplicada e Computacional e do
Departamento de Matemática em especial ao Prof. Dr. José Roberto Nogueira, pelos conhecimentos
transmitidos.
Aos sete companheiros que junto a mim acreditaram e formaram a primeira turma de Pós-
Graduação em Matemática Aplicada e Computacional da FCT-UNESP : Diego (um amigo nos
momentos difíceis e alegres nesses dois anos de disciplinas e pesquisa), Marluce, Marluci, Marilaine,
Marcelo, Claudio e Danilo (in memoriam) e aos amigos da segunda turma Larissa, Camila, Clóvis,
Merejolli, Verri, Pedro, Patrícia e de um modo especial à Cristiane e Tatiane pela amizade e excelente
companhia no período nal de meu trabalho. Aos preciosos amigos Flávio, Thaís e Alex que apesar
da distância física estiveram sempre presentes em minha vida acadêmica e também ao amigo Renan
por sempre me ajudar quando precisei. Enm, à todos que direta ou indiretamente contribuíram
para o desenvolvimento deste trabalho.
Aos funcionários da Seção de Pós-Graduação, em especial à Erynat pelo auxílio prestado no
decorrer deste curso de mestrado.
À Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP pelo apoio nanceiro.
Não serei o poeta de um mundo caduco.
Também não cantarei o mundo futuro.
Estou preso à vida e olho meus companheiros.
Estão taciturnos mas nutrem grandes esperanças.
Entre eles, considero a enorme realidade.
O presente é tão grande, não nos afastemos.
Não nos afastemos muito, vamos de mãos dadas...
Carlos Drummond de Andrade
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(
ψ1 (n+ 1) ψ2 (n+ 1)
ψ1 (n) ψ2 (n)
)
.
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‖Ψ(n)‖ =+*+ 7$(+ &$-,.<$ (+ )8,+.<$ () +,7$;+-$*)&" >+*+ )&7,(+* $ 4*)&4'5)%7$ ()
‖ME,ω (n)‖3 ()?%'5$&
γω,± (E) = limn→∞
1
nln ‖ME,ω (n)‖ .
9 )@'&7A%4'+ ()&&) -'5'7) &):,) ($ *)&,-7+($ +6+'@$ 0;)B+ C11D2"
!"#!$% &'()' !"#$%&'(&#)*+&$%&', -.#. /.0. E ∈ C1 &23$%&4 ΩE ⊆ Ω & γ (E) ∈ R 0&
4505 6"& µ (ΩE) = 1 & 7.#. /.0. ω ∈ ΩE1 γω,± (E) &23$%&4 & $85 3)".3$ . γ (E)1 3$%5 91
γω,+ (E) = γω,− (E) = γ (E) .
E %F5)*$ γ (E) G 4H+5+($ &275&'%& 0& :;.7"'5<"
!"#!$% &'((' =$/&>&0&/, ?"75'@. 6"& 7.#. .>)"4 E ∈ C1 γ (E) > 0A B'%851 7.#.
%505 ω ∈ ΩE1 &23$%&4 $5>"CD&$ ψ+d , ψ
−d 0& Hωψ = Eψ 0& 4505 6"& ψ±d 0&/.&4 &275*
'&'/3.>4&'%& &4 ±∞1 #&$7&/%3<.4&'%&1 /54 %.2. −γ (E)A E>94 03$$51 %50. $5>"C85 6"&
9 >3'&.#4&'%& 3'0&7&'0&'%& /54 ψ+d #&$7&/%3<.4&'%&1 ψ−d , /#&$/& &275'&'/3.>4&'%& 7.#.
+∞ #&$7&/%3<.4&'%&1 −∞, /54 %.2. γ (E)A
9 ()5$%&7*+.<$ ()&7) *)&,-7+($ =$() &)* )%4$%7*+(+ )5 C!I3 !JD" 9&&'53 %$ 4+&$ ($
)@=$)%7) () KL+=,%$; =$&'7';$3 7)5$& ,5+ 4$5=*))%&<$ 4$5=-)7+ ($ 4$5=$*7+5)%7$
+&&'%7M7'4$ (+& &$-,./)& %$ '%?%'7$"
9 7)$*'+ () N$7+%' )&7+6)-)4) ,5+ -':+.<$ )%7*) $ )@=$)%7) () KL+=,%$; ) $ )&=)47*$
+6&$-,7+5)%7) 4$%7O%,$" P)?%+
A = E ∈ R : γ (E) = 0 .
E Q)4H$ )&&)%4'+- Sess
() ,5 4$%B,%7$ S ⊆ R G ()?%'($ =$*
Sess
= E ∈ R : ℓ ((E − ǫ, E + ǫ) ∩ S) > 0 ∀ǫ > 0 ,
$%() ℓ ()%$7+ + 5)('(+ () K)6)&:,)" R5 =+*7'4,-+*3 Sess
=∅ =+*+ 7$($ 4$%B,%7$ S ()
5)('(+ () K)6)&:,) S)*$"
! !"#$%&' () '"*+!,'+*- ,* - .+/,012*+ 3,
!"#!$% &'(&' !"#$$%&'"()*%+,('-$. Σac = Aess
/
"#$# # %&'()*+$#,-( %&*+& $&*./+#%( 0&1# 2!34 5 4 5678 9 :$;<='( $&*./+#%( :(%& *&$
&)>()+$#%( &' 25!78
!"#!$% &'()' +,('-$. 01 ," 2,(1-3$'$" Vω "4, '21*$56$3," 1 '"")717 ",71-(1 )7
-871*, 9-$(, 61 :';,*1"< 1-(4, ℓ (A) = 0/ =7 2'*($3);'*< Σac =∅/
?&**& '(%(4 *& >()*=%&$#$'(* .'# @#'A/=# &$B;%=># %& (:&$#%($&* %& C>D$E%=)B&$
(Hω)ω∈Ω4 %&F)=%# :($ G!8!H4 *&)%( '=)='#/ & (* :(+&)>=#=* Vω #:&$=;%=>(* #**.'=)%( .'
)I'&$( F)=+( %& 0#/($&*4 &)+-( :&/(* $&*./+#%(* #)+&$=($&* +&'(* J.&
σ (Hω) = σp (Hω) ∪ σsc (Hω) .
K**='4 J.#)%( &*+.%#'(* ( +=:( &*:&>+$#/ %& Hω4 :$&>=*#'(* #:&)#* #)#/=*#$ ( &*:&>+$(
:()+.#/ & ( &*:&>+$( *=)B./#$ >()+A).( Hω8
! "#$%&'&#'()*% +,'-'&'./% * 01&/()*% 2/ S1
C&1# A .' #/@#L&+(8 M'# *.L*+=+.=,-( ζ N .'# #:/=>#,-( ζ : A → A∗8 K *.L*+=O
+.=,-( ζ :(**.= &<+&)*P&* )#+.$#=*4 :($ >()>#+&)#,-(4 # A∗ & AN4 *&)%( ζ (b1 · · · bn) =
ζ (b1) · · · ζ (bn) & ζ (b1, b2 · · · ) = ζ (b1) ζ (b2) · · · 8?=Q&'(* J.& ζ N :$='=+=0# *& &<=*+& k ∈ N +#/ J.& :#$# >#%# a ∈ A4 ζk (a) >()+N'
+(%(* (* *A'L(/(* %& A8 R<&':/(* =':($+#)+&* %& *.L*+=+.=,P&* :$='=+=0#* *-( %#%#* :($S
=H T=L()#>>=S
ζf (a) = ab ζf (b) = a
==H ?.:/=>#,-( %& "&$A(%(S
ζdp (a) = ab ζdp (b) = aa
===H U=)V$=# )-(O"=*(+S
ζbp (a) = ab ζbp (b) = aaa
=0H WD.&OX($*&S
ζtm (a) = ab ζtm (b) = ba
0H Y.%=)OCD#:=$(S
ζrs (a) = ab ζrs (b) = ac
! ! "#$"%&%#&'()" *+&,&%&-." ) +/%.'()" 0. S1 !
ζrs (c) = db ζrs (d) = dc
!"#$%&' ()*+) !" #$%&'()*" +$ #&,#-*-&*./0 1 &! 20(-0 340 5#$ $4*#-*67 +$ &!"
#&,#-*-&*./0 ζ $! AN8 0& #$9"8 &! $:$!$(-0 u ∈ AN
-": %&$ ζ (u) = u;
" #$%&'()*%+ ,# -. /0)'0 1$0 ,# -.+ &-2&'%'-%3405 &026# -. +78+2#'0 A5 9 +&&#:-6+,+
/#7+& &#:-%)'#& *0),%3;#& <=#>+ ?!@ABC
• D$%&'# a ∈ A '+7 E-# ζ (a) *0.#3+ *0. + 7#'6+ aF
• limn→∞ |ζn (a)| = +∞5 ∀a ∈ AF
":06+ =+.0& -&+6 #&&+& &#E-()*%+&5 *0. =+706#& )-.96%*0&5 /+6+ *0)&'6-%6 /0'#)*%+%& /+6+
0& 0/#6+,06#& ,# G*H6I,%):#6 <JFJBF
K0,#.0& #&*6#=#6 u = limn→∞ ζn (a)5 &#),0 u +7:-. /0)'0 1$0 ,+ &-2&'%'-%340 /6%.%L
'%=+ ζF D&'#),#),0 u +62%'6+6%+.#)'# /+6+ + #&E-#6,+5 &#),0 u ∈ AZ# ,#1)%),0 Ω *0.0 0
*0)>-)'0 ,0& /0)'0& ,# +*-.-7+340 ,0& '6+)&7+,+,0& ,# u5 *0. + 6#&/#*'%=+ *0)=#6:()*%+
/0)'-+75
Ω =
ω ∈ AZ : ω = lim
ni→∞T ni u
.
M%N#.0& E-# Ω 9 0 <&:: ,# ω #. AZ5 &#),0 T 0 0/#6+,06 #<*=- &026# AZ
5 ,#1)%,0 /06
(T u)n = un+15 u ∈ AZF K06 ,#1)%3405 Ω 9 -. &-2*0)>-)'0 8#*H+,0 ,# AZ
0 E-+7 9
%)=+6%+)'# &026# T F O &%&'#.+ ,%)P.%*0 (Ω, T ) 9 *H+.+,0 #*#-$!" +*(>!*)0 +$ #&,#-*-&*./0
"##0)*"+0 " ζF G#),0 ζ /6%.%'%=+5 #&'# &%&'#.+ /0&&-% /60/6%#,+,#& 2#. *0)H#*%,+& < =#>+
#. ?!@AB5 0 E-+7 9 -)%*+.#)'# #6:Q,%*05 %&'0 95 #$%&'# &0.#)'# -.+ .#,%,+ ,# /602+2%7%,+,#
#6:Q,%*+ %)=+6%+)'#F R+.29.5 '#.0& E-# #&'# &%&'#.+ 9 .%)%.+75 0- &#>+5 + Q62%'+ ,# '0,0
#7#.#)'0 ω ∈ Ω 9 ,#)&0 #. ΩF S%)+7.#)'#5 '0,+ /+7+=6+ 0*066#),0 #. +7:-. ω ∈ Ω
0*066# #. '0,0 ω ∈ ΩF
T0'#.0& E-# + σ−+7:#26+ ,# U06#7 9 :#6+,+ /#70& *0)>-)'0& *%7V),6%*0&
[b0 . . . bl−1][m,m+l−1] = ω ∈ Ω : ωm+i = bi, 0 ≤ i ≤ l − 1 ,
&#),0 m ∈ Z, l ≥ 1, bi ∈ A, 0 ≤ i ≤ l− 1F R#.0& E-# + W)%*+ .#,%,+ #6:Q,%*+ ,# U06#7
µ &026# Ω &+'%&8+N
µ(
[b0 . . . bl−1][m,m+l−1]
)
= d (b0 . . . bl−1) ,
0),# d (b0 . . . bl−1) 9 + 86#E-()*%+ ,+ /+7+=6+ b0 . . . bl−1 #. u5 %&'0 95
d (b0 . . . bl−1) = limn→∞
1
n♯ j ≤ n : uj . . . uj+l−1 = b0 . . . bl−1 ,
! !"#$%&' () '"*+!,'+*- ,* - .+/,012*+ 3,
" #$"% & '()*+( (',+-,")(.,( */'-,-0"1 2'3/%4(.5/ "+6-,+"+-")(.,( $)" 7$.89/ f : A→ R
( 5(:.-.5/ g (ω) = f (ω0); /6,()/' /' */,(.3-"-' Vω (n) = f (ωn)1
<('$)-5")(.,(; */5()/' 5-=(+ #$( '$*/.5/ " '$6',-,$-89/ ζ *+-)-,-0"; " '(#$>.3-"
5( '$6',-,$-89/ u .9/ & *(+-?5-3" ./ :."%; ( " 7$.89/ f "''$)( ./ )@.-)/ 5/-' 0"%/+('1
2.,9/ " 7")@%-" -.5$=-5" 5( /*(+"5/+(' (Hω)ω∈Ω & (+A?5-3" ( )-.-)"%1
BA/+"; 0")/' (',$5"+ " 3/.',+$89/ 5( $)" 7")@%-" (+A?5-3" 5( /*(+"5/+(' 5( C34+DE
5-.A(+ FG(:.-89/ 2.6H; "''/3-"5/' "/' */,(.3-"-' Vω A(+"5/' */+ +/,"8I(' ." 3-+3$.7(+>.3-"
S11 J) )/5(%/ 5( +/,"89/ ." 3-+3$.7(+>.3-" & *"+")(,+-="5/ */+ ,+>' *"+K)(,+/'L
1 $) .M)(+/ 5( +/,"89/ -++"3-/."% α ∈ (0, 1)N
O1 $) 3/)*+-)(.,/ 5( -.,(+0"%/ β ∈ (0, 1)N
P1 $)" 3/.',".,( 5( "3%/*")(.,/ .9/ .$%" λ ∈ R1
2Q-',() 5$"' )".(-+"' *"+" ('3/%4(+ Ω, T, g1 B *+-)(-+" )".(-+" '(A$( 5/ 2Q()*%/ 2.3;
'(.5/ Ω = S1 ≃ [0, 1); ( " "*%-3"89/ Tαω = ω+α (mod 1) *"+" "%A$) -++"3-/."% α ∈ (0, 1);
/ #$"% (Ω, Tα) & (',+-,")(.,( (+A?5-3/1 G(:.-)/' " 7$.89/ g 3/)/ g (ω) = λ ·χ[1−β,1) (ω);
( (',(' A(+") /' */,(.3-"-'
Vω (n) = g (T nω) = λ · χ[1−β,1) (αn+ ω mod 1) .
R$,+" */''-6-%-5"5( 5( /6,(+ $)" 7")@%-" 5( */,(.3-"-' & 3/.'-5(+"+ " '(#$>.3-" vα,β(n) =
χ[1−β,1) (αn mod 1); n ∈ N1 G(:.-.5/ / !"" Ω = Ωα,β */+
Ωα,β =
ω ∈ 0, 1Z : ω = limT nivα,β, ni →∞
,
,()/' #$( / '-',()" 5-.K)-3/ (Ωα,β, T ) & (',+-,")(.,( (+A?5-3/; 3/) $)" M.-3" )(5-5"
(+A?5-3" µ 5"5" '/6+( 3/.S$.,/' 3-%@.5+-3/' *(%" 7+(#$>.3-" 5"' +('*(3,-0"' *"%"0+"' TO!U1
B 7$.89/ g #$( A(+" /' */,(.3-"-' .(',( 3"'/; & 5"5" */+ g (ω) = f (ω0); /.5( f (0) = 0 (
f (1) = λ; '(.5/ A = 0, 11V()/' #$( ")6"' "' )".(-+"' 5( /6,(+ /' */,(.3-"-' 5( (Hω)ω∈Ω 3/++('*/.5(.,(' "/'
*"+K)(,+/' α, β, λ -.5$= W 7")@%-"' 5( /*(+"5/+(' 5( C34+D5-.A(+ (+A?5-3"' ( )-.-)"-'1
X/ 3"'/ 5( α = β ('3+(0()/' vα ( Ωα; ( /6,()/' / '-',()" 5-.K)-3/ (Ωα, T ) 3/) /'
+('$%,".,(' */,(.3-"-' #$!%&'()*+1 B%A/ (Q,+()")(.,( -.,(+(''".,( & #$( ." '$6',-,$-89/
5( Y-6/."33- / !"" "''/3-"5/ & (#$-0"%(.,( "/ !"" C,$+)-"./ 3/++('*/.5(.,( " α =√5−12
;
'(.5/ a 7→ 1 ( b 7→ 01 G(''( )/5/; "' 3/++('*/.5(.,(' 7")@%-"' 5( /*(+"5/+(' '9/
(#$-0"%(.,('1
!"! #$%&'#()* +,#(* !
!" #$%&'()*+ ,-()+
" #$%&'( )( *%+,-$./( )$ 0(%)(. 1234(5(& &,+$%$ * 6.#$&/6+*7'( )$ $&/%,/,%*& %$8$2
/6/6#*& .( 8(/$.56*4 $ * 46-6/*7'( )( /%*7( )*& -*/%69$& )$ /%*.&:$%;.56* *&&(56*)*& *(
(8$%*)(% )$<.6)( $- =1>1?>
@( 5*&( )(& -()$4(& A/,%-6*.(& (, -()$4(& +$%*)(& 8(% &,3&/6/,67B$& 8%6-6/6#*&C *
46-6/*7'( )( /%*7( )*& -*/%69$& )$ /%*.&:$%;.56* 8()$ &$% 6.#$&/6+*)* */%*#D& )( $&/,)( )$
,- &6&/$-* )6.E-65(C 5F*-*)( *8465*7'( /%*7(C ( G,*4 D 6.),96)( 8(% $&/%,/,%*& %$8$/6/62
#*& )( 5(%%$&8(.)$./$ 8(/$.56*4> H&&*& $&/%,/,%*& $&/'( 8%$&$./$& .(& 8(/$.56*6& +$%*)(&
8(% &,3&/6/,67B$& 8%6-6/6#*& $ &,* 8%$&$.7* .( 5*&( A/,%-6*.( 8()$ &$% $I636)* ,&*.)( *
$I8*.&'( $- :%*7B$& 5(./6.,*)*& )( .J-$%( )$ %(/*7'( 6%%*56(.*4 *&&(56*)( *( 8(/$.56*4>
@$&/$ $&/,)(C #*-(& &$+,6% (& 8%(5$)6-$./(& )$ *.*46&*% ,-* &$G,;.56* )$ 8*4*#%*&
+$%*)(%*& (3$)$5$.)( *& %$4*7B$& %$5,%&6#*&C 5(.&6)$%*% *& -*/%69$& )$ /%*.&:$%;.56* *&&(2
56*)*& K 8*4*#%*& <.6/*& $- #$9 )$ &$G,;.56*& 6.<.6/*&C $&/$.)$% *& %$4*7B$& %$5,%&6#*&
)*& 8*4*#%*& 8*%* ( .L#$4 )*& -*/%69$& )$ /%*.&:$%;.56* $C 5(.&$G,$./$-$./$C 8*&&*% $&&*&
%$4*7B$& 8*%* (& /%*7(& )$&&*& -*/%69$&>
M*%* (& -()$4(& *&&(56*)(& *& &,3&/6/,67B$& 8%6-6/6#*&C /$-(& G,$ 8*%* 5*)* E <I(C
5(.&6)$%$-(&C 8(% *3,&( )$ .(/*7'(C * *8465*7'( ME : A→ SL (2,R)C )$<.6)* 8(%
ME (a) =
(
E − f (a) −11 0
)
.
N*-(& )$.(/*% 8$4( -$&-( &L-3(4( * *8465*7'( ME : An → SL (2,R)C )$<.6)* 8(%
ME (ω) =ME (an) · · ·ME (a1) , =1>O?
&$.)( ω = (a1 · · · an)> " *8465*7'( ME 8$%-6/$ 6./%(),96% ,-* *7'( 6.),96)* )$ ζ &(3%$
( 5(.P,./( 6-*+$- )$ ME 5(- ζ (ME (ω)) ≡ME (ζω)C )(.)$ ζn (ME (ω)) =ME (ζn(ω))>
Q$&&$ -()(C )$#6)( * $G,*7'( =1>O? /$-(& G,$ * *7'( )$ ζ )$<.$ ,- &6&/$-* )6.E-65(
&(3%$ SL (2,R)|A|C )$&)$ G,$ ME (ζn(a)) 8()$ &$% $I8%$&&*)( 5(-( 8%(),/( )$ -*/%69$&
ME (ζn−1(a))C a ∈ A>
" *.R46&$ )$&&$ &6&/$-* )6.E-65(C $- 8%6.5L86( 8()$%6* 8%(),96% /()* * 6.:(%-*7'( )$2
&$P*)* &(3%$ ( $&8$5/%( )( (8$%*)(% =1>1?> H./%$/*./(C .* 8%R/65* 8()$ &$% $I/%$-*-$./$
5(-8465*)( /%*3*4F*% 5(- $&/$ &6&/$-* )6%$/*-$./$C $ $I6&/$ * #*./*+$- )$ &$ 8*&&*% 8*%*
,- .(#( &6&/$-* )6.E-65( 5(- 3*&$ .(& /%*7(& )*& -*/%69$& ME (ζn(a))>
Q$<.6-(&C 8*%* 5*)* ω ∈ A∗C xE (ω) = tr (ME (ω))> @*/,%*4-$./$C 8()$-(& $&5%$#$%
xnE (ω) = tr (ME (ζnω))C $ (3#6*-$./$ 8()$-(& $&/$.)$% * *7'( )$ ζC 5(-( ζ(
xn−1E (ω))
=
! !"#$%&' () '"*+!,'+*- ,* - .+/,012*+ 3,
xnE (ω)" #$ %&'(&'$) *%++( ,%- &.$ %/0+'% 12( %/34%++.$ 02%*0('( *% ζ(
xn−1E (a))
) 5$2$
61&7.$ *% xn−1E (a)) $1 +%8() ( 4%(90-(7.$ *%++( (7.$ *% ζ 5$2$ 12 +0+'%2( *0&:205$ %2
R|A|" ;$4<2) =1(&*$ 0++$ &.$ 6$4 3$++>,%9 +%234% 3$*%2$+ %&5$&'4(4 12 +1?5$&81&'$
@&0'$ A ⊂ A∗ '(9 =1% 3(4( '$*$ ω ∈ A) xnE (ω) 3$*% +%4 %/34%++(*$ 5$2$ 12( 61&7.$ *%
xn−1E (ω)) 5$2 ω ∈ A) $1 +%8() ( 4%(90-(7.$ *% ζ 5$2$ 12 +0+'%2( *0&:205$ +$?4% R|A|"
A(9 +0+'%2( *0&:205$ < 5B(2(*$ *% (3905(7.$ '4(7$"
!"#$%& '()*( ! "#$! %# $&'$()(&)*+! %, -)'!.#"")/ (,0!$ 1&,
ζnf (a) = ζn−1f (ab) = ζn−1f (a) ζn−1f (b) ,
ζnf (b) = ζn−1f (a) .
2#3# $)045)6"#3 # .!(#*+!/ %,.!(#0!$
xn = tr(
ME
(
ζnf (a)))
yn = tr(
ME
(
ζnf (b)))
zn = tr(
ME
(
ζnf (a))
ME
(
ζnf (b)))
.
7,0!$ 1&,
xn = tr(
ME
(
ζn−1f (a))
ME
(
ζn−1f (b)))
= zn−1
,
yn = tr(
ME
(
ζn−1f (a)))
= xn−1.
89!3#/ &$#.%! 1&,
tr (M1M2M1M3) = tr (M1M2) tr (M1M3) + tr (M2M3)− tr (M2) tr (M3) ,
$,.%! Mi ∈ SL (2,R) , i = 1, 2, 3/ !'(,0!$ 1&,
zn = tr(
ME
(
ζn−1f (a))
ME
(
ζn−1f (b))
ME
(
ζn−1f (a)))
= xn−1xn − xn−2.
,$(, ,:,045! .+! ; 43,")$! #045)#3 ! #5<#',(! => 1&, zn = xn+1? 8$$)0/ (,0!$ # 3,"&3$+!
,1&)@#5,.(, xn = zn−1 = xn−1xn−2 − xn−3/ ,.@!5@,.%! $!0,.(, xA@#3)>@,)$?
!"! #$%&'#()* +,#(* !
!"#$%& '()*( ! "#$! %# $&'$()(&)*+! %&,-)"#*+! %. ,./0!%!1 (.2!$ 3&.
ζndp (a) = ζn−1dp (ab) = ζn−1dp (a) ζn−1dp (b) ,
ζndp (b) = ζn−1dp (aa) = ζn−1dp (a) ζn−1dp (a) .
4#/# $)2,-)5"#/ # 6!(#*+!1 %.6!(#2!$
xn = tr(
ME
(
ζndp(a)))
yn = tr(
ME
(
ζndp(b)))
7$#6%! ! 8.!/.2# %. 9#-.:;<#2)-(!61 !'(.2!$ 3&.
yn = tr(
ME
(
ζn−1dp (a))2)
= tr(
ME
(
ζn−1dp (a)))
tr(
ME
(
ζn−1dp (a)))
− tr (I)
= x2n−1 − 2.
=>!/#1 &$#6%! # )%.6()%#%.
tr (M1M2) = tr (M1) tr (M2)− tr(
M1M−12
)
,
$.6%! Mi ∈ SL (2,R) , i = 1, 21 !'(.2!$ 3&.
xn = tr(
ME
(
ζn−1f (a))
ME
(
ζn−1f (b)))
= tr(
ME
(
ζn−1f (a)))
tr(
ME
(
ζn−1f (b)))
− 2
= xn−1yn−1 − 2.
?!>!1 6+! @ ,/.")$! #2,-)#/ ! #-A#'.(! $.6%! # /."&/$+! .3&)B#-.6(. xn = x3n−1−2xn−1−21
.6B!-B.6%! $!2.6(. x;B#/)CB.)$D
"#$# % &#'% (%' )%*+,&-#-' .+$#(%' )%$ $%*#/0+' ,# &-$&1,2+$3,&-#4 5+6#7%' #8.17#'
)$%)$-+(#(+' )#$# % E&-- 9*1$7-#,%: ;%,'-(+$+ # +<)#,'=% +7 2$#/0+' &%,*-,1#(#' (+
α ∈ (0, 1) -$$#&-%,#84
α =1
a1 +1
a2+1
a3+···
>?:@A
'+,(% an ∈ N 1,-+,*+ (+*+$7-,#(%':
! !"#$%&' () '"*+!,'+*- ,* - .+/,012*+ 3,
" #$%&'()#*+& %#,(&-#. #//&,(#0# αn = pnqn
1 023-(0# $2.#/ %2.#*42/ %2,5%/(6#/7
p0 = 0, p1 = 1, pn = anpn−1 + pn−2
q0 = 1, q1 = a1, qn = anqn−1 + qn−2. 89:;<
=23-()&/ #/ $#.#6%#/ sn /&>%2 & #.?#>2@& 0, 1 $&%
s−1 = 1, s0 = 0, s1 = sa1−10 s−1, sn = sann−1sn−2, n ≥ 2. 89:A<
B) $#%@(,5.#%C # $#.#6%# sn $&//5( ,&)$%()2-@& qnC n ≥ 2: =2//2 )&0&C $&02)&/ 023-(%
5)# /2D5E-,(# (-3-(@# 5-(.#@2%#. 0# /2F5(-@2 ?&%)#C
u = uα = limn→∞
sn.
" /2F5(-@2 $%&$&/(*+& %2.#,(&-# uαC &5 /2G#C #/ $#.#6%#/ snC ,&) & !"" H@5%)(#-& Ωα:
!"#"$%&'" ()*+) vα #$%&#'&( )*#* 1, 2, 3, . . . +(',+'-$ +(. uα/
" 02)&-/@%#*+& 02/@2 %2/5.@#0& 2-,&-@%#I/2 2) J9K:
!"#"$%&'" ()*,) 0*#* +*-* n ≥ 21 snsn+1 = sn+1san−1n−1 sn−2sn−1/
-./"0$1!2&'")" 02)&-/@%#*+& /2F52 0(%2@#)2-@2 0# 2D5#*+& 89:A<: =2 ?#@&C
snsn+1 = snsan+1n sn−1 = san+1
n snsn−1 = san+1n sann−1sn−2sn−1
= sn+1san−1n−1 sn−2sn−1.
B/,%262-0& sn = s1n . . . sqnn C 023-()&/ #/ )#@%(L2/ 02 @%#/?2%E-,(# ME (sn) ,&%%2/$&-I
02-@2/ #/ $#.#6%#/ sn $&%
ME (sn) =
(
E − λsqnn −11 0
)
× · · · ×(
E − λs1n −11 0
)
.
"F&%#C 5)# D52/@+& -#@5%#. $#%# /2% #-#.(/#0# 1 D5#. # %2.#*+& 0& 2/$2,@%& ,&)
& ,&-G5-@& 0#/ 2-2%F(#/ &-02 & @%#*& 0#/ )#@%(L2/ 02 @%#-/?2%E-,(# 1 .()(@#0&: M2//2
/2-@(0&C N2.(//#%0 2 &5@%&/ 02)&-/@%#%#) 2) J9K 5)# (F5#.0#02 2-@%2 2//2/ ,&-G5-@&/
$#%# &/ )&02.&/ F2%#0&/ $&% $&@2-,(#(/ H@5%)(#-&/: O#(/ 2'$.(,(@#)2-@2C
!"! #$%&'#()* +,#(* !
!"#"$%&'" ()*+) !"# (Hω)ω∈Ω $%# &#%'()# !*+,-).# -! /0!*#-/*!1 -! .2*3-)4+!* 5$*6
%)#4/17 8459/ 0#*# λ, α ./**!10/4-!45!1: !;)15! $%# ./415#45! Cλ 5#( <$!
σ (Hω) = E : |tr (ME (sn))| ≤ Cλ ∀n .
=/5#4-/ <$! 1! -!>4)%/1 xn = tr (ME (sn)): yn = tr (ME (sn−1)) ! zn ? tr (ME (snsn−1)):
/@5!%/1 <$!
max|xn|, |yn|, |zn| ≤ Cλ.
"#$%&' & ()&* +,- #./%$&'01 20'0 #3 1#4&5#3 033#6%04#3 03 37.3/%/7%89&3 2'%1%/%$03:
%;657%;4# <%.#;066%: 4725%608=# 4& 2&'>#4#: .%;?'%# ;=#@A%3#/ & B)7&@C#'3&: 710 60'06@
/&'%*08=# 40 025%608=# /'08# 6#1 # &32&6/'#D E;/'&/0;/#: ;#/01#3 F7& ;=# 3& #./G1 0
5%1%/08=# 403 H'.%/03 4& &;&'I%03 4# &32&6/'# &1 3&7 6#;/&J/# I&'05: 103 3%1 0 5%1%/0@
8=# 20'0 710 37.3&F7K;6%0D L&3/& 3&;/%4#: 0 025%608=# /'08# 033#6%040 0 1#4&5#3 4&
37.3/%/7%8=# 2'%1%/%$0 &J%.% 02&;03 710 0;05#I%0 20'6%05 M A'#2#3%8=# 2.19D
!"#$%&'
!"#$%&'() *% +(!*(&
! "#$%&'()*! +' ,*#+*( !-* &.)*+*! '!!'(/0"0! 1"#" '2/3%0# * '!1'/)#* 1*()%"3
+*! *1'#"+*#'! +' 4/5#6+0($'# 78 +* )01* 97:7;: <'!)' /"1=)%3* >"&*! "("30!"# "3$%&"!
>'#!?'! +0!/#')"! ' %&" >'#!-* /*()=(%" +*! "#$%&'()*! +' ,*#+*(@ !'(+* A%' (*!!*
*BC')0>* /*(!0!)' '& '!)%+"# "! +'&*(!)#"D?'! +*! "#$%&'()*! +' ,*#+*( EFB3*/*! ' GF
B3*/*! 9H'*#'&"! 7:E ' 7:G; ' )"&B.& %&" >'#!-* +0!/#')" +* "#$%&'()* +' ,*#+*(@
1#'!'()' (" #'I'#J(/0" KLM:
<" !'$%(+" !'D-* '!)%+"&*! * /"!* /*()=(%* *(+' "("30!"&*! * #'!%3)"+* +' 8"&"(0N
' 4)*3O K7PM !*B#' %&" >'#!-* /*()=(%" &"0! $'#"3 +* "#$%&'()* +' ,*#+*(@ 1"#" 1*F
)'(/0"0! +' ,*#+*( $'('#"30O"+*! 98'Q(0D-* 3.4;@ * A%"3 #'/%1'#" *! #'!%3)"+*! *#0$0("0!
+*! 1*)'(/0"0! +' ,*#+*(: H"3 #'!%3)"+* $"#"()' A%' *! *1'#"+*#'! +' 4/5#6+0($'# 78
"!!*/0"+*! " 1*)'(/0"0! +' ,*#+*( $'('#"30O"+*! (-* 1*!!%'& "%)*>"3*#'! 9H'*#'&" 1.4;:
!" #$%&'$& ()&*%$+,&
<'!)" !'D-* '!)%+"&*! " +'&*(!)#"D-* +' #'!%3)"+*! '!!'(/0"0! +* (*!!* 1#*C')*@ /*&*
*! "#$%&'()*! +' ,*#+*( EFB3*/*! ' GFB3*/*!@ !'(+* A%' '!!'! +*0! &.)*+*! 1*+'& !'#
%)030O"+*! 1"#" '2/3%0# "%)*>"3*#'! +* '!1'/)#* +* *1'#"+*# +' 4/5#6+0($'# +0!/#')* 78@
* 1#0&'0#* /*& #'1')0D-* +' +*0! B3*/*! +* 1*)'(/0"3@ C%()"&'()' /*& 30&0)"D-* +* )#"D*
+"! &")#0O'! +' )#"(!I'#J(/0"@ ' * !'$%(+* /*& #'1')0D-* +' )#J! B3*/*! +* 1*)'(/0"3 9K7RM;:
S"#" " "130/"D-* +'!)'! &.)*+*! >"&*! Q2"# %& '3'&'()* +" I"&=30" '#$T+0/" (Hω)ω∈Ω@ '
ER
!"! #$%&'$& ()&*%$+,& !
"#$# %#&# "'()*%+#, ,+-+(#&' V : Z→ R #..'%+#-'. ' '")$#&'$ &) /%0$1&+*2)$
(Hψ) (n) = ψ (n+ 1) + ψ (n− 1) + V (n)ψ (n) , 345!6
) # )78#9:' &'. #8(';#,'$).
(Hψ) (n) = ψ (n+ 1) + ψ (n− 1) + V (n)ψ (n) = Eψ (n) , 345 6
'*&) E ∈ C5 <) -'&' #*=,'2' #' 78) >'+ >)+(' "#$# 8-# >#-?,+# )$2@&+%# &) '")$#&'$).
&) /%0$1&+*2)$A &)B*+-'. #. -#($+C). &) ($#*.>)$D*%+# ME (n) = ME,ω (n) 3ω ∈ Ω BE'65
F)-'. 78) ψ G .',89:' &) (Hψ) (n) = Eψ (n) .)A ) .'-)*() .)A Ψ G .',89:' &) Ψ(n) =
ME (n)Ψ (0)A &'*&) Ψ(n) =
(
ψ (n+ 1)
ψ (n)
)
A "#$# ('&' n ∈ Z5
!"# $%&% !"# ME (n) # $#%&'( )! %&#*+,!&-*.'# #++/.'#)# #/ /0!&#)/& H ! ./*+')!&!
Ψ : N −→ C21$# +!21-*.'#3 !
‖Ψ(2n)‖ ≥ 1
2/1 ‖tr (ME (n))Ψ (n)‖ ≥ 1
20#&# %/)/ n ∈ N,
!*%4/
max (‖Ψ(2n)‖ , ‖Ψ(n)‖) ≥ 1
2min
(
1,1
|tr (ME (n))|
)
.
'!"()*+,#-.(%H#-'. .8"'$ 78) "#$# ('&' n ∈ NA
‖Ψ(2n)‖ ≥ 1
2'8 ‖tr (ME (n))Ψ (n)‖ ≥ 1
2.
I*(:'
max (‖Ψ(2n)‖ , ‖Ψ(n)‖) ≥ 1
2
'8
max (‖Ψ(2n)‖ , ‖Ψ(n)‖) ≥ 1
2 |tr (ME (n))| .
J'$(#*('A
max (‖Ψ(2n)‖ , ‖Ψ(n)‖) ≥ 1
2min
(
1,1
|tr (ME (n))|
)
.
K2'$#A $)%'$&#*&' ' F)'$)-# 1.2 3#$28-)*('. &) L'$&'* MN,'%'.A )*8*%+#&' *#
O*($'&89:'6A )- 78) %'*.+&)$#-'. ME (n) # -#($+C &) ($#*.>)$D*%+#A BE#&'. 8- "'()*%+#,
!"#$%&' () !*+%,-.$'/ 0- +'*0'.
V ! "# !$%&'&( E ∈ C &$$)%*&+)$ &) ),!(&+)( H +!-.*+) ,)( /01234 ).+! 5!#)$ & (!,!5*67)
+! 8')%)$ +) ,)5!.%*&' ,&(& "#& $!9":.%*& nk −→ ∞ ! "#& '*#*5&67) .) 5(&6) +&$
#&5(*;!$ +! 5(&.$<!(:.%*&4 !.57) )85!#)$ $)8 !$$&$ =*,>5!$!$4 9"! E .7) ? "# &"5)@&')(
+! H4 )" $!A&4 ) !$,!%5() ,).5"&' +! H ? @&;*)1 B!A&#)$ & +!#).$5(&67) +!$$! (!$"'5&+)1
!"#$%&'()*#+,-!#'!"( 1.2. C*D! "# !$%&'&( E ∈ C ! "#& <".67) '*#*5&+& /,)5!.E
%*&'3 V : Z −→ R4 &$$)%*&+)$ &) ),!(&+)( H1 B&#)$ $",)( 9"! E ? "# &"5)@&')( +! H4
)" $!A&4 !D*$5! ψ ∈ l2 (Z)− 0 5&' 9"!
(Hψ) (n) = Eψ (n) .
F$%(!@!.+) !$5& !9"&67) !# 5!(#)$ +&$ #&5(*;!$ +! 5(&.$<!(:.%*&4 5!#)$ 9"!
Ψ(nk) =ME (nk)Ψ (0) .
G)#) ME (nk) ? "#& #&5(*; 9"&+(&+& +! )(+!# 24 ,)+!#)$ !$%(!@!( $!" ,)'*.H#*)
%&(&%5!(I$5*%)J
p (λ) = λ2 − tr (ME (nk))λ+ 1.
F.57)4 ,!') K!)(!#& +! G&L'!LEM&#*'5).4
ME (nk)2 − tr (ME (nk))ME (nk) + I = 0. /0103
NO)(&4 $",).+) 9"! V (j) = V (j + nk)4 1 ≤ j ≤ nk4 +!@*+) & +!-.*67) +&$ #&5(*;!$ +!
5(&.$<!(:.%*&4
ME (nk)2 =ME (2nk) ,
! $"8$5*5"*.+) .& !9"&67) /0103 )85!#)$
ME (2nk)− tr (ME (nk))ME (nk) + I = 0.
G).$*+!(&.+) ) @!5)( *.*%*&' Ψ(0)4 %)# ‖Ψ(0)‖ = 14 ! &,'*%&.+)E) .& !9"&67) &%*#&
5!#)$ 9"!
Ψ(0) = tr (ME (nk))Ψ (nk)−Ψ(2nk) .
P!$$! #)+)4 ,!'& +!$*O"&'+&+! 5(*&.O"'&( 5!#)$
1 ≤ ‖tr (ME (nk))Ψ (nk)‖+ ‖Ψ(2nk)‖ .
!"! #$%&'$& ()&*%$+,& !
"##$%&
‖tr (ME (nk))Ψ (nk)‖ ≥1
2'( ‖Ψ(2nk)‖ ≥
1
2.
)* tr (ME (nk)) = 0& *+,-' ‖Ψ(2nk)‖ ≥ 12& ' .(* /'+,012$3 ' 41,' 2* ψ ∈ l2 (Z)5 )*
tr (ME (nk)) 6= 0& *+,-' 6*7' 8*%1 3.1 ,*%'#
max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖) ≥1
2min
(
1,1
|tr (ME (nk))|
)
.
9'0 :$6;,*#* |tr (ME (nk))| ≤ C& '+2* 1 ≤ C <∞5 <+,-'
min
(
1,1
|tr (ME (nk))|
)
≥ 1
C.
8'='&
max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖) ≥1
2C.
>1#& .(1+2' nk −→ ∞ ,*%'# .(* ‖Ψ(nk)‖ −→ 0& 6'$# ψ ∈ l2 (Z)& * /'%' ' %?@$%' A
(%1 167$/1B-' /'+,C+(1 #*=(* .(*
1
2C≤ lim
nk→∞max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖) = 0,
' .(* A (%1 /'+,012$B-'5
9'0,1+,'& /'+/7(C%'# .(* E +-' 6'2* #*0 (% 1(,'D17'0 6101 H& * 6*7' %*#%' 1E#(02'
+*+:(%1 #'7(B-' 21 *.(1B-' Hψ = Eψ 6'2* ,*+2*0 1 3*0' *% +∞5
<+,0*,1+,'& *% 17=(%1# #$,(1BF*# 6'2* #*0 (%1 ,10*41 /'%67$/121 7$%$,10 ' ,01B'
21 %1,0$3 2* ,01+#4*0G+/$1 ME (n)5 <+,-'& 6'2*H#* ,*+,10 *+/'+,010 (%1 0*6*,$B-' 2'#
D17'0*# 2' 6',*+/$17 V I *#.(*021& 2'+2* ,*%'# ' 10=(%*+,' 2* J'02'+ !HE7'/'#5 K*L1%'#
1 2*%'+#,01B-' 2' M*'0*%1 1.3 N*+(+/$12' +1 O+,0'2(B-'P5
!"#$%&'()*#+,-!#'!"( 1.3.Q* %'2' 1+?7'=' 1 2*%'+#,01B-' 2' M*'0*%1 1.2& D1%'#
#(6'0 .(* E A (% 1(,'D17'0 2* H& '( #*L1& *@$#,* ψ ∈ l2 (Z)− 0 ,17 .(*
(Hψ) (n) = Eψ (n) .
<#/0*D*+2' *#,1 *.(1B-' *% ,*0%'# 21# %1,0$3*# 2* ,01+#4*0G+/$1& ,*%'# .(*
Ψ(nk) =ME (nk)Ψ (0) .
! !"#$%&' () !*+%,-.$'/ 0- +'*0'.
"# $%&$# '()$# *+% ,# -%$(,&.)#/0( -( .%()%$# #,.%)1()2 (3.%$(&
ME (2nk)− tr (ME (nk))ME (nk) + I = 0. 456!7
8%9# :1;<.%&% -# )%;%.1/0( -(& 39(=(& &(3 ( ;(.%,=1#9 V 2 .%$(& *+%ME (nk)ME (−nk) = I6
>;91=#,-( Ψ(−nk) ,# %*+#/0( 456!72 )%&+9.# *+%
tr (ME (nk))Ψ (0) = Ψ (nk) + Ψ (−nk) .
?($#,-( ‖Ψ(0)‖ = 12 ;%9# -%&1@+#9-#-% .)1#,@+9#) .%$(&
|tr (ME (nk))| ≤ ‖Ψ(nk)‖+ ‖Ψ(−nk)‖ . 456A7
8#)# (& B#9()%& -% nk %$ *+% |tr (ME (nk))| ≥ 12 ;() 456A7
‖Ψ(nk)‖+ ‖Ψ(−nk)‖ ≥ 1,
%,.0(
‖Ψ(nk)‖ ≥1
2(+ ‖Ψ(−nk)‖ ≥
1
2.
>@()#2 ;#)# (& B#9()%& -% nk %$ *+% |tr (ME (nk))| ≤ 12 #;91=#,-( Ψ(0) ,# %*+#/0(
456!7 &%@+% *+%
1 = ‖Ψ(0)‖ ≤ ‖Ψ(2nk)‖+ ‖Ψ(nk)‖ ,
9(@(
‖Ψ(nk)‖ ≥1
2(+ ‖Ψ(2nk)‖ ≥
1
2.
8().#,.(2 ;#)# .(-( nk2
max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖ , ‖Ψ(−nk)‖) ≥1
2.
>&&1$2 *+#,-( nk −→ ∞ .%$(& *+% ‖Ψ(nk)‖ −→ 02 ;(1& ψ ∈ l2 (Z)6 C($( ,( .%()%$#
#,.%)1() 1&.( ,(& 9%B# # +$# =(,.)#-1/0(2 ;(1&
1
2≤ lim
nk→∞max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖ , ‖Ψ(−nk)‖) = 0.
"%&&# '()$#2 =(,=9+D$(& *+% E ,0( ;(-% &%) +$ #+.(B#9() ;#)# H2 % *+% ,%,:+$#
&(9+/0( -# %*+#/0( Hψ = Eψ ;(-% .%,-%) # E%)( %$ ±∞6
!"! #$%&'$& ()&*%$+,& !
"#$%&' ()*&+$, -$ .)$%)+& &/&01$ 2+& $2.%& ()%,3$ 40,5%).& 4$, &%#2+)-.$, 4) 6$%7
4$- 89: ;&%& $ $;)%&4$% H 4)<-04$ ;$% =>?@A' $-4) 5$-,04)%&7,) 2+& &;%$10+&B3$ C0+0.&4&
) ;)%0D405& 4$ ;$.)-50&C V ' ,)-4$ ;$,,E()C )15C20% $, &2.$(&C$%), &,,$50&4$, &$ $;)%&4$%
H?
!"#!$% &'(' !"#$ V (n) ! Vm (n)% &#'# m ∈ N% (!)*+,-.#( /.$.0#1#( (23'! Z 4.(02 5%
n ∈ Z67 *&2,8#$2(
97 Vm &!'.:1.-#% -2$ &!';212 Tm →∞<
=7 supm,n |Vm (n)| <∞<
>7 sup|n|≤2Tm |Vm (n)− V (n)| ≤ m−Tm7
?,0@2 )*#/)*!' (2/*A@2 ψ 6= 0 1# !)*#A@2 12( #*02B#/2'!( =>? A (#0.C#D
lim sup|n|→∞
|ψ (n+ 1)|2 + |ψ (n)|2
|ψ (1)|2 + |ψ (0)|2≥ 1
4.
F$+$ 2+& &;C05&B3$ 4$ G)$%)+& 3.2 .)+$, H2) $ +$4)C$ "C+$,.7I&.J0)2' ;&%& 5)%.&,
%)&C0K&BL), 4$ ;$.)-50&C' ;$,,20 ),;)5.%$ ;$-.2&C (&K0$ =()*& M)B3$ N?>A?
)!$% &'&' !"# M ∈ SL (R, 2) ! x *$ B!02' *,.0E'.27 ?,0@2
max(
‖Mx‖ ,∥
∥M2x∥
∥ ,∥
∥M−1x∥
∥ ,∥
∥M2x∥
∥
)
≥ 1
2.
*!$"+,-#%./"'O)C$ G)$%)+& 4) F&C)P7Q&+0C.$-' .)+$, H2)
M2 − tr (M)M + I2 = 0. =>?RA
",,0+' ,) |tr (M)| ≥ 1' ,)*& K = 1|tr(M)| ? S-.3$' &;C05&-4$ M−1
) $ ().$% 2-0.T%0$ x -&
)H2&B3$ =>?RA' .)+$,
x = K(
Mx+M−1x)
.
U),,) +$4$'
1 = ‖x‖ ≤ |K|(
‖Mx‖+∥
∥M−1x∥
∥
)
=>?9A
≤ ‖Mx‖+∥
∥M−1x∥
∥ . =>?VA
W$#$' ‖Mx‖ ≥ 12$2 ‖M−1x‖ ≥ 1
2?
! !"#$%&' () !*+%,-.$'/ 0- +'*0'.
"#$%&' () |tr (M)| < 1' &*+,-&./$ x .& )01&23$ 456!7 8)9$( 01)
x = tr (M)Mx−M2x.
:$.()01).8)9).8)'
1 = ‖x‖ < ‖Mx‖+∥
∥M2x∥
∥ .
;$#$' ‖Mx‖ ≥ 12$1 ‖M2x‖ ≥ 1
26 <$%8&.8$'
max(
‖Mx‖ ,∥
∥M2x∥
∥ ,∥
∥M−1x∥
∥ ,∥
∥M−2x∥
∥
)
≥ 1
2.
!"#$%&'()*#+,-!#'!"( 3.2. =)>&9 ψ ) ψm ($+12?)( /& )01&23$ /) &18$@&+$%)( Hψ =
Eψ' &9A&( -$9 &( 9)(9&( -$./,2?)( ,.,-,&,( (&8,(B&C)./$ |ψ (1)|2 + |ψ (0)|2 = 16 :$9$
.$( -&($( &.8)%,$%)(' -$.(,/)%)9$( &( ()01D.-,&(
Ψ(n) =
(
ψ (n+ 1)
ψ (n)
)
, Ψm (n) =
(
ψm (n+ 1)
ψm (n)
)
.
E.83$' 18,+,C&./$ &( 9&8%,C)( /) 8%&.(B)%D.-,&' *$/)9$( )(-%)@)% & )01&23$
Ψ(n) =ME (n)Ψ (0) .
F) 9$/$ &.G+$#$' 8)9$( 01) Ψm (n) = MmE (n)Ψ (0)6 <$% ,./123$ ($A%) n H *$((I@)+
9$(8%&% 01)
‖Ψm (n)−Ψ(n)‖ ≤ |n|C |n−1| supn,m|V (n)− Vm (n)‖ ‖Ψ(0)‖ ,
()./$ C = max|i| (‖TE (i)‖ , ‖TmE (i)‖)J |i| = 1, . . . , n6 ;$#$' 1(&./$ & K,*L8)() 4,,7 8)9$(
sup|n|≤2Tm
‖Ψm (n)−Ψ(n)‖ ≤ sup|n|≤2Tm
|n|C |n−1|m−Tm
≤ sup|n|≤2Tm
|n|eC|n|m−Tm
= 2Tme2CTmm−Tm .
E.83$'
maxa=±1,±2
‖Ψm (aTm)−Ψ(aTm)‖ → 0, 456M7
!"! #$%&'( )(*+,*-. !
"#$%&' m→∞( )*'+$, -./$ -.+0'&010&$&. &' -'2.%10$/ Vm . -./' 3.4$ 3.3
maxa=±1,±2
‖Ψm (aTm)‖ ≥1
2. 56(789
:%2;', -./$< ."#$=>.< 56(?9 . 56(789, 1'%1/#@4'< "#.
lim sup|n|→∞
|ψ (n)|2 + |ψ (n+ 1)|2 ≥ lim sup|n|→∞
maxa=±1,±2
‖Ψ(aTm)‖2 ≥1
4.
!" #$%&'( )(*+,*-.
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'-.+$&'+ &. E1F+G&0%*.+ &$ H'+4$
H = − d2
dx2+ V (x) , 56(779
$2#$%&' <'I+. L2 (R), 1'4 #4 -'2.%10$/ V : R → R /'1$/4.%2. 0%2.*+JB./ 5 V ∈ L1loc9(
K$+201#/$+4.%2., .<2$4'< 0%2.+.<<$&'< .4 -'2.%10$0< &$ H'+4$
V (x) = V1 (x) + V2 (αx+ θ) , 56(7 9
'%&. V1 . V2 <;' /'1$/4.%2. 0%2.*+JB.0<, -'<<#.4 -.+@'&' 7 . α, θ ∈ [0, 1)(
E. α = pqL +$10'%$/, .%2;' ' -'2.%10$/ V 2.4 -.+@'&' q . H -'<<#0 .<-.12+' $I<'/#2$M
4.%2. 1'%2@%#' -#+'( E. α L 0++$10'%$/, .%2;' ' -'2.%10$/ L "#$<.M-.+0N&01', . ' .<-.12+'
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E1F+G&0%*.+ H( T. 4$%.0+$ <.4./F$%2., B$4'< .<2#&$+ #4$ *.%.+$/0C$=;' &' +.<#/2$&'
&. D'+&'%, -$+$ '-.+$&'+.< H &$ H'+4$ 56(779, 1'4 -'2.%10$0< V &.U%0&'< $I$0P'(
!"#$%&' ()*) !"#$%& '(# V ) ($ *%+#,-!./ 0# 1%20%, 3#,#2./!".0% &# V ∈ L1loc,unif (R)4
! !"#$%&' () !*+%,-.$'/ 0- +'*0'.
!"# $%
‖V ‖1,unif = supx∈R
∫ x+1
x
|V (x)| dx <∞,
& &' !"&( )#"&*+ , ! V m% -& )&./#-#! Tm →∞% ", ! 01&
limm→∞
eCTm∫ 2Tm
−Tm|V (x)− V m (x)| dx = 0, "#$%#&
!&*-# C 1(, +#*!",*"& ,).#). ,-,2
'()*)+,-.,/ .010 20.,-34)( 1, 50*10- 6 7+ 20.,-34)( 1, 50*10- 8,-,*)(49)10/ 204:
;7)-10 m→∞
eCTm∫ 2Tm
−Tm|V (x)− V m (x)| dx ≤ eCTm3CTm
mTm→ 0.
!"# $%&% 31)#*4, 01& V ∈ L1loc,unif (R) & !&5, Hψ = Eψ +#( ψ ∈ L2 (R)2 6*"7#%
|ψ (x)|2 + |ψ′ (x)|2 → 0,
01,*-# |x| → ∞2
< 1,+0-:.*)=>0 1,:., (,+) :,87, 10: .,0*,+): '$#$% , '$#$ 1, ?@#A$
'0+0 -0 3):0 3(B::430 ? #A/ C)+0: +0:.*)* ;7, :, V 6 7+ 20.,-34)( 1, 50*10- 8,-,D
*)(49)10/ ,-.>0 2)*) .01) ,-,*84) E ): :0(7=E,: 1,
−ψ′′ (x) + V (x)ψ (x) = Eψ (x) "#$%@&
->0 .,-1,+ ) 9,*0 ;7)-10 |x| → ∞/ 07 :,F)/
|ψ (xm)|2 + |ψ′ (xm)|2 ≥ D
2)*) 7+) 30-:.)-., D > 0 , 7+) :,;7G-34) (xm)m∈N ) ;7)( 0H,1,3, |xm| → ∞ ;7)-10
m → ∞$ I-.>0/ ->0 ,J4:.,+ :0(7=E,: ,+ L2 (R)/ 204: :, ψ ∈ L2 (R) 4+2(43)*4) 2,(0
K,+) 3.5 ;7, |ψ (xm)|2 + |ψ′ (xm)|2 → 0/ ;7)-10 |xm| → ∞$
L4J0: 104: 20.,-34)4: W1 ∈ L1loc,unif (R)/ W2 ∈ L1
loc (R) , 7+) ,-,*84) E/ 30-:41,*,+0:
): :0(7=E,: ψ1, ψ2M
−ψ′′1 (x) +W1 (x)ψ1 (x) = Eψ1 (x)
−ψ′′2 (x) +W2 (x)ψ2 (x) = Eψ2 (x) ,
!"! #$%&'( )(*+,*-. !
"#$ %& "#'()*+,& )')")%)& ψ1 (0) = ψ2 (0)- ψ′1 (0) = ψ′2 (0)-
|ψ1 (0)|2 + |ψ′1 (0)|2 = |ψ2 (0)|2 + |ψ′2 (0)|2 = 1.
!"# $%&% !"#$% C = C(
‖W1 − E‖1,unif)
$&' ()% *&+& $,-, x. $%/,#
∥
∥
∥
∥
∥
(
ψ1 (x)
ψ′1 (x)
)
−(
ψ2 (x)
ψ′2 (x)
)∥
∥
∥
∥
∥
≤ CeC|x|∫ max(0,x)
min(0,x)
|W1 (t)−W2 (t)| |ψ2 (t)| dt. ./0123
'!"()*+,#-.(%4#'&)(,5,$#& x ≥ 0 .%& $#()6"%*+,& 7%5% x < 0 &8# 9:;)%&30 <,$#&
(
ψ1 (x)− ψ2 (x)
ψ′1 (x)− ψ′2 (x)
)
=
∫ x
0
(
ψ′1 (t)− ψ′2 (t)
(W1 (t)− E)ψ1 (t)− (W2 (t)− E)ψ2 (t)
)
dt =
=
∫ x
0
(
0
(W1 (t)−W2 (t))ψ2 (t)
)
dt+
∫ x
0
(
ψ′1 (t)− ψ′2 (t)
(W1 (t)− E) (ψ1 (t)− ψ2 (t))
)
dt =
=
∫ x
0
(
0
(W1 (t)−W2 (t))ψ2 (t)
)
dt+
∫ x
0
(
0 1
W1 (t)− E 0
)(
ψ1 (t)− ψ2 (t)
ψ′1 (t)− ψ′2 (t)
)
dt.
=#>#-
∥
∥
∥
∥
∥
(
ψ1 (x)− ψ2 (x)
ψ′1 (x)− ψ′2 (x)
)∥
∥
∥
∥
∥
≤∫ x
0
|W1 (t)−W2 (t)| |ψ2 (t)| dt
+
∫ x
0
∥
∥
∥
∥
∥
(
0 1
W1 (t)− E 0
)∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
(
ψ1 (t)− ψ2 (t)
ψ′1 (t)− ψ′2 (t)
)∥
∥
∥
∥
∥
dt.
?,@# @,$% (, A5#'B%@@ CDEF #:G,$#&
∥
∥
∥
∥
∥
(
ψ1 (x)
ψ′1 (x)
)
−(
ψ2 (x)
ψ′2 (x)
)∥
∥
∥
∥
∥
≤ e
∫ x
0
∥
∥
∥
∥
(
0 1
W1 (t)− E 0
)∥
∥
∥
∥
dt∫ x
0
|W1 (t)−W2 (t)| |ψ2 (t)| dt.
H,&&, $#(#- "#'"@IJ$#& KI, ,L)&G, I$ ;%@#5 C = C(
‖W1 − E‖1,unif)
G%@ KI, % 5,@%*8#
./0123 M &%G)&N,)G%0
O# @,$% %")$%- #:&,5;%$#& KI, 7#(,$#& "#'G5#@%5 % ()N,5,'*% (, &#@I*+,& ,$ G,5P
$#& (%& "#'()*+,& (, I$% )'G,>5%@ ,';#@;,'(# % ()N,5,'*% (#& 7#G,'")%)&0 QIG5# N%G#
)$7#5G%'G, 7%5% % (,$#'&G5%*8# (# <,#5,$% 1.4 M KI, 7%5% 7#G,'")%)& 7,5)9()"#&- G,$#&
"#'R,")$,'G# &#:5, % '#5$% (# ;,G#5 &#@I*8# (ψ (x) , ψ′ (x))T ,$ (,G,5$)'%(#& 7#'G#& x-
! !"#$%&' () !*+%,-.$'/ 0- +'*0'.
"#$%& '"(& )#'(& $& *#+, , "#-.'/0
!"# $%&% !"#$%& W !' "#()$*+&, *#' ")-.#/# p ) E !'& )$)-0+& &-1+(-2-+&3 4$(5#
(#/& 6#,!75# /)
ψ′′ (x) +W (x)ψ (x) = Eψ (x) , 1 2345
$#-'&,+8&/& $# 6)$(+/# 9!) |ψ (0)|2 + |ψ′ (0)|2 = 1: 6&(+6;&8 & )6(+'&(+<&
max
(∥
∥
∥
∥
∥
(
ψ (−p)ψ′ (−p)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (p)
ψ′ (p)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (2p)
ψ′ (2p)
)∥
∥
∥
∥
∥
)
≥ 1
2.
'!"()*+,#-.(%6,+&" 7&$"'%#/,/ 8.# ψ 9 .+, "&*.:;& %# 1 23452 <,/, x, y ∈ R= x < y=
'$(/&%.>'+&" ?,/, & 7,"& 7&$(@$.& , +,(/'> %# (/,$")#/A$7', ME (x, y)= ",('"),>#$%& 8.#
ME (x, y)
(
ψ (x)
ψ′ (x)
)
=
(
ψ (y)
ψ′ (y)
)
.
B#+&" 8.# #"", +,(/'> %#?#$%# "&+#$(# %, #$#/-', E # %& ?&(#$7',* "&C/# & '$(#/D,*&
(x, y)2 E$(;&= 7&+& W ?&"".' ?#/@&%& p= (#+&"
ME (−p, 0) =ME (0, p) =ME (p, 2p) :=ME. 1 23F5
G-&/,= .",$%& & (#&/#+, %# H,*#IJK,+'*(&$= "#$%& det (ME) = 1 1D#L, , &C"#/D,:;& ,?M"
#"", %#+&$"(/,:;&5= (#+&" 8.#
M2E − tr (ME)ME + I = 0. 1 23N5
O# |tr (ME)| ≤ 1= ,?*'7,+&" , #8.,:;& ,7'+, $& D#(&/ (ψ (0) , ψ′ (0))T = # ?#*, /#*,:;& 1 23F5
"#-.# 8.#
(
ψ (2p)
ψ′ (2p)
)
− tr (ME)
(
ψ (p)
ψ′ (p)
)
+
(
ψ (0)
ψ′ (0)
)
= 0.
P#""# +&%&= 7&+& #"(,+&" 7&$"'%#/,$%& |ψ (0)|2 + |ψ′ (0)|2 = 1= &C(#+&"
1 =
∥
∥
∥
∥
∥
(
ψ (0)
ψ′ (0)
)∥
∥
∥
∥
∥
≤∥
∥
∥
∥
∥
(
ψ (p)
ψ′ (p)
)∥
∥
∥
∥
∥
+
∥
∥
∥
∥
∥
(
ψ (2p)
ψ′ (2p)
)∥
∥
∥
∥
∥
.
Q&-&=
max
(∥
∥
∥
∥
∥
(
ψ (p)
ψ′ (p)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (2p)
ψ′ (2p)
)∥
∥
∥
∥
∥
)
≥ 1
2.
!"! #$%&'( )(*+,*-. !
"# |tr (ME)| > 1$ %# &'(#)*' +#&#,-'(.# '/ 0'+/ '(.#*)/*$ '1,)0'&/+ ' #23'45/ 6 7!89 (/
:#./* (ψ (−p) , ψ′ (−p))T $ # 1#,' *#,'45/ 6 7!;9
(
ψ (p)
ψ′ (p)
)
− tr (ME)
(
ψ (0)
ψ′ (0)
)
+
(
ψ (−p)ψ′ (−p)
)
= 0.
</=/$ 0/&/ |u (0)|2 + |u′ (0)|2 = 1$ .#&/+ 23#
1 < |tr (ME)|∥
∥
∥
∥
∥
(
ψ (0)
ψ′ (0)
)∥
∥
∥
∥
∥
≤∥
∥
∥
∥
∥
(
ψ (p)
ψ′ (p)
)∥
∥
∥
∥
∥
+
∥
∥
∥
∥
∥
(
ψ (−p)ψ′ (−p)
)∥
∥
∥
∥
∥
.
>/*.'(./$
max
(∥
∥
∥
∥
∥
(
ψ (p)
ψ′ (p)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (−p)ψ′ (−p)
)∥
∥
∥
∥
∥
)
≥ 1
2.
!"#$%&'()* ?/&/ (/ 0'+/ %)+0*#./$ +# 0/(+)%#*'*&/+ 23# ψ, ϕ +5/ +/,34@#+ %' #23'45/
6 7!A9$ 1'*' x, y ∈ R # 1'*' 0'%' #(#*=)' BC' E$ .#&/+ 23# '+ &'.*)D#+ %# .*'(+E#*F(0)'
ME (x, y) %'%'+ 1/*
ME (x, y)
(
ψ (x)
ψ′ (x)
)
=
(
ψ (y)
ψ′ (y)
)
e ME (x, y)
(
ϕ (x)
ϕ′ (x)
)
=
(
ϕ (y)
ϕ′ (y)
)
,
1/++3#& 0/,3('+
(
ψ (y)
ψ′ (y)
)
#
(
ϕ (y)
ϕ′ (y)
)
$ %#+%# 23# '+ 0/(%)4@#+ )()0)')+ %# ψ, ϕ #& x
+#G'&H
(
ψ (x)
ψ′ (x)
)
=
(
1
0
)
,
(
ϕ (x)
ϕ′ (x)
)
=
(
0
1
)
.
I#&/+ 23# / J*/(+K)'(/ %'+ +/,34@#+ ψ, ϕ %' #23'45/ 6 7!A9 L %#B()%/ 1/*
w (ψ, ϕ) = ψ (y)ϕ′ (y) + ϕ (y)ψ′ (y) ,
# #+.' #C1*#++5/ )(%#1#(%# %# yM ,/=/ /N.#&/+ 23# det (ME (x, y)) = 17
O)(',&#(.#$ :'&/+ E'D#* ' %#&/(+.*'45/ %/ I#/*#&' 1.4 #(3(0)'%/ (' )(.*/%345/7
!"#$%&'()*#+,-!#'!"( 1.4. "#G' V 3& 1/.#(0)', %# P/*%/( =#(#*',)D'%/ # +#G'&
V m'1*/C)&'(.#+ %# 1#*Q/%/ Tm$ +'.)+E'D#(%/ ' *#,'45/ 6 7! 97 O)C'(%/ m$ '1,)0'(%/ /
! !"#$%&' () !*+%,-.$'/ 0- +'*0'.
"#$% 3.6 &'$ W1 = V # W2 = V m(#$')
∥
∥
∥
∥
∥
(
ψ (x)
ψ′ (x)
)
−(
ψm (x)
ψ′m (x)
)∥
∥
∥
∥
∥
≤ C1eC1|x|
∫ max(0,x)
min(0,x)
|V (t)− V m (t)| |ψm (t)| dt, * +,-.
'/0# ψ 1 )'2345' 0# −ψ′′ (x) + V (x)ψ (x) = Eu (x) # ψm 1 )'2345' 0# −ψ′′m (x) +
V m (x)ψm (x) = Eψm (x)6 )#/0' %$7%) %) )'2348#) /'9$%2:;%0%) /% '9:<#$ # '7#0#&#$
%) $#)$%) &'/0:48#) 0# &'/('9/'+ =#$') >#2% #?3%45' * +, . ?3# ‖V m‖1,unif 1 2:$:(%0'
#$ m+ @/(5'6 3$% )#<3/0% %>2:&%45' 0' "#$% 3.6 &'$ W1 = V m# W2 = 06 /'(%/0' ?3#
% &'/)(%/(# #$ * +,A. 0#>#/0# )'$#/(# 0% /'9$% )'79# L1loc,unif 0# W1 − E6 (#$') ?3#
∥
∥
∥
∥
∥
(
ψm (x)
ψ′m (x)
)
−(
ψ0 (x)
ψ′0 (x)
)∥
∥
∥
∥
∥
≤ C2eC2|x|
∫ max(0,x)
min(0,x)
|V m (t)| |ψ0 (t)| dt,
0'/0# C2 /5' 0#>#/0# 0# m # ψ0 1 3$% )'2345' /'9$%2:;%0% 0# −ψ′′0 = Eψ0+ B'(%/0'
?3# ψ0 1 2:$:(%0' #C>'/#/&:%2$#/(#6 (#$') ?3#
|ψm (x)| ≤ C3eC3|x|
* +!D.
&'$ C3 :/0#>#/0#/(# 0# m *$%:) 0#(%2E#) #$ FG H.+ I#))# $'0'6 )37)(:(3:/0' * +!D. #$
* +,-. >%9% ('0' x6 &'$ −Tm ≤ x ≤ 2Tm6 (#$')
∥
∥
∥
∥
∥
(
ψ (x)
ψ′ (x)
)
−(
ψm (x)
ψ′m (x)
)∥
∥
∥
∥
∥
≤ CeC|Tm|∫ 2Tm
−Tm|V (t)− V m (t)| dt,
'/0# C = max 2 (C1 + C3) , C1C3+ J#2% #?3%45' * +, . (#$') ?3# #C:)(# m0 (%2 ?3# >%9%
('0' m ≥ m06 (#$')
∥
∥
∥
∥
∥
(
ψ (x)
ψ′ (x)
)
−(
ψm (x)
ψ′m (x)
)∥
∥
∥
∥
∥
≤ 1
4* +!,.
>%9% ('0' x6 &'$ −Tm ≤ x ≤ 2Tm+ K'$' V m(#$ >#9L'0' Tm6 )#<3# >#2' "#$% 3.7 ?3#
max
(∥
∥
∥
∥
∥
(
ψm (−Tm)ψ′m (−Tm)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψm (Tm)
ψ′m (Tm)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψm (2Tm)
ψ′m (2Tm)
)∥
∥
∥
∥
∥
)
≥ 1
2.
I#))# $'0'6 >#2% 0#):<3%20%0# %&:$% # >'9 * +!,. '7(#$') ?3#
max
(∥
∥
∥
∥
∥
(
ψ (−Tm)ψ′ (−Tm)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (Tm)
ψ′ (Tm)
)∥
∥
∥
∥
∥
,
∥
∥
∥
∥
∥
(
ψ (2Tm)
ψ′ (2Tm)
)∥
∥
∥
∥
∥
)
≥ 1
16.
!"! #$%&'( )(*+,*-.
!"# $"%&'()*%+', +'-"& .)'
|ψ (xm)|2 + |ψ′ (xm)|2 ≥1
4
/0#0 )-0 &'.)1%$*0 xm ∈ −Tm, Tm, 2Tm 0 .)02 "3'4'$' |xm| → ∞ .)0%4" m→∞5
6'&&' -"4", /'2" 7'-0 3.5 $"%$2)8-"& .)' " "/'#04"# H %9" /"&&)* 0)+":02"#'&5
;- %"&&" +#0302<" '&+0-"& *%+'#'&&04"& '- /"+'%$*0*& 40 ="#-0 > 5?@A, &'%4" α ∈ [0, 1)
)- %B-'#" *##0$*"%025 C'-"& .)' α /"4' &'# 4'$"-/"&+" '- =#0DE'& $"%+*%)040& $"-"
'- >@5FA ' &'# 3'- 0/#"G*-04" /"# %B-'#"& #0$*"%0*& .)' &0+*&=0H'- 0& #'20DE'& #'$)#&*:0&
>@5IA5 6'&&' -"4", &' $"%&*4'#0#-"& 0& 0/#"G*-0DE'& V m4' /'#8"4" qm 4'J%*40& $"-"
V m (x) = V1 (x) + V2 (xαm + θ) , > 5@@A
"3+'-"& *-'4*0+0-'%+' " &'()*%+' $"#"2K#*" /0#0 " C'"#'-0 1.4L
!"!#$"%! &'(' !"#$%&'#( )!* *+,(-* !'& .#$(-&$-* C -&/ )!*
limm→∞
eCqm∫ 2qm
−qm|V2 (xα + θ)− V2 (xαm + θ)| dx = 0.
0$-1# V 23&3# "#4 > 5?@A5 6 !' "#-*$.,&/ 3* 7#43#$ 7*$*4&/,8&3# * H 23*9$,3# "#4
> 5??A5 "#((!, *("*.-4# "#$-!&/ :&8,#;
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!"#$%&'()
!"#! $%&'#()* +%,*" !"#(-%. %)/(,%" %&)0$%12!" -*" %./(,!3#*" -! 4*.-*3 %3%)0"%-*"
3* $%&'#()* %3#!.0*.5 6)7, -0""*8 !"#(-%.!,*" &.*&.0!-%-!" #%0" $*,* 9!"&!$#.* $*,
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3)+ ,+& J243& +#-4-& 42*&.4&2%# µ> -#K24+)$
Ωc = ω ∈ Ω : σp (Hω) = ∅ .
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+,'-'.$%&' ()3) B(2%+C) '(# G (n)7 n ∈ N7 &9% ,%+;(+.%& -# D%1#0 .)!& '(#
E? lim supn→∞G (n) ⊆ Ωc ,
F? lim supµ (G (n)) > 0.
! !"#$%&' () !"&* !+,-.
!"#$% µ (Ωc) = 1&
!"#$%&'()*#+" #$%&'()*+ %$,&$ &%)-*+ .&$
µ
(
lim supn→∞
G (n)
)
≥ lim supn→∞
µ (G (n)) .
/$') 0123($%$ (1) ($4+% .&$
µ (Ωc) ≥ µ
(
lim supn→∞
G (n)
)
,
'+,+ µ (Ωc) > 05 6-(7+8 2$') 1-9)#1:-;1) *$ Ωc8 ;+-;'&1<%$ .&$ µ (Ωc) = 15
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0#%#8 "'5) @.#"#(+A9# 4.2 6#$65!B0#( &!' Ωc C !0 Gδ %'$(#>
D( (+0'*.+)( %# *+"# 1#.%#$ ,#.)0 '$6#$*.)%)( $# 6)(# E+4#$)66+ FGHI8 $# 6)(# *!.3
0+)$# /'.)5 F2I ' ").) !0) 65)((' %' 0#%'5#( %' (!4(*+*!+A9# %!"5+6)A9# %' "'.B#%# F7I>
J)04C0 %'+K)0#( 6#0# #4('.<)A9# &!' # *.)4)5:# F2HI ,#.$'6' !0 0C*#%# ").) %'0#$(3
*.). ) )!(L$6+) /'$C.+6) %' )!*#<)5#.'( '(*!%)$%# '(*.!*!.)( ")5+$%.M0+6)( $#( "#*'$6+)+(>
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6)")- %' '(*)4'5'6'. .'(!5*)%#( &>*>" #! !$+,#.0'(>
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6'0#( !()$%# # )./!0'$*# %' 1#.%#$ 7345#6#(> ?'R$)
G (n) = ω ∈ Ω : Vω (k − n) = Vω (k) = Vω (k + n) , 1 ≤ k ≤ n .
S4<+)0'$*'8 #( G (n) (9# 6#$O!$*#( %' T#.'58 OU &!' '5'( (9# !$+V'( R$+*)( %' 6#$O!$*#(
6+5B$%.+6#(>
,'#-#%.)*# /+0+ !"#$%& '!(
lim supn→∞
µ (G (n)) > 0.
)$*+#, µ (Ωc) = 1-
!"#$%&'()*#+ @'5# )./!0'$*# %' 1#.%#$ 7345#6#(8 #4*'0#( lim supG (n) ⊆ Ωc> ?)B8
"'5) @.#"#(+A9# 4.38 6#$65!B0#( &!' µ (Ωc) = 1>
J)04C0 *'0#( &!' µ (G (n)) "#%' ('. '(*+0)%# )*.)<C( %)( ,.'&!L$6+)( %#( 6!4#(8
%'<+%# ) '&!)A9#
µ(
ω ∈ Ω : ωm · · ·ωm+|v|−1 = v)
= d (v) .
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&!' 'K+(*' !0) &!).*) "#*L$6+) v4 #6#..'$%# '0 u *)5 &!' |v| = n> N$*9# µ (G (n)) ≥nd (v4)> N0 ").*+6!5).8 ) :+"Y*'(' %) @.#"#(+A9# 4.5 C ()*+(,'+*) (' '$6#$*.).0#( !0)
6#$(*)$*' B > 0 ' !0) ('&!L$6+) %' ")5)<.)( vk 6#0 |vk| = nk → ∞8 &!)$%# k → ∞8
*)5 &!' ").) *#%# k8 v4k ∈ Fψ ' d (v4k) ≥ Bnk>
! !"#$%&' () !"&* !+,-.
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0#)%9#$ )% -#'*:;# &* .(-.7&+%-<&.(*=
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8-) +9 4+):4$)%1)9 an %" );/"%93+ )# 50"2<)9 4+%1$%-"="9 =) α 9"1$95">)#
lim supn→∞
an ≥ 4.
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µ (Ωc) = 17
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*.(0* .#0# $%67%C L%E&* 8*-* n ∈ N % C <∞1
G′
(n,C) = ω ∈ Ω : Vω (k) = Vω (k + n) , 1 ≤ k ≤ n, |tr (ME,ω (n))| ≤ C ∀E ∈ Σ .
"#H*0%&'% '%0#$ M7% #$ G′
(n,C) $;# 7&(N%$ E&('*$ )% .#&I7&'#$ .(9F&)-(.#$ %1 8#-'*&'#1
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lim supn→∞
µ(
G′
(n,C))
> 0.
?%13+@ µ (Ωc) = 17
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.7?# v3 #.#--%&)# %0 u '*9 M7% |v| = n % |tr (ME,ω (n))| ≤ C, ∀E ∈ ΣC @&';#
µ(
G′
(n,C))
≥ nd(
v3)
.
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0/D*(8'/ +/#%E
2!$% &'34' $%&' v (n) = χ[1−α) (nα mod 1)/ 0,)1-
+: v (n) = [(n+ 1)α]− [nα]4 ∀n ∈ Z/
++: v (qn + k) = v (k)4 1 ≤ k < qn+1 − 14 n 6= −1/
+++: v (−n) = v (n− 1)/
! !"#$%&' () !"&* !+,-.
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0'1 '*21' 34.'+
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0
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'-.% · .%-'24 4 5412% 614,"'-71"4/ 0'1 8&+ 2%&'( )*%
[(n+ 1)α]− [nα] = 1⇔ ∃m ∈ N; m+ 1− α ≤ nα < m+ 1, (%-.' m = [nα] .
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v (n) = [(n+ 1)α]− [nα] , ∀n ∈ Z.
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'D2%&'( )*%
v (qn + k) = [(qn + k + 1)α]− [(qn + k)α]
= [(k + 1)α + qnα− pn]− [kα + qnα− pn]
= [(k + 1)α]− [kα] = v (k) .
"""# 0%3' "2%& "# (%:*% )*%
v (−n) = [(−n+ 1)α]− [−nα] = [α− nα] + [nα] = [nα]− [(n− 1)α] = v (n− 1) .
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" "#$%&'"% $()*'&'" (&)+(,&- " &%.*/$0)" '$ 1"%'"0 2345"6"(7 .&%&0)+0'" 8*$ #&%& α, λ
9:"7 Ωc 6= ∅; <* &+0'&7 #$5" =$/& 4.10 )$/"( & %$#$)+>?" 0" #")$06+&5 & @'+%$+)&A $ &
@$(8*$%'&A7 5"." (&)+(,&- " &%.*/$0)" '$ 1"%'"0 B345"6"(7 '"0'$ Ωc 6= ∅; C"%)&0)" #$5&
C%"#"(+>?" 4.47 *)+5+-&0'" " &%.*/$0)" '$ 1"%."0 2345"6"( "* B345"6"( D #"((EF$5 "4)$%
%$(*5)&'"( .$0D%+6"( ("4%$ & $:65*(?" '$ &*)"F&5"%$(;
C"()$%+"%/$0)$ &" %$(*5)&'" "4)+'" &6+/& $/ G2H7 I&/+0&.& G2JH $()$0'$* "( %$(*5)&'"(
'$ K$L5"0 $ C$)%+)+( G!JH7 "0'$ 6"0(+'$%"* " 6"0M*0)"
G (n) = ω ∈ Ωα : Vω (k − qn) = Vω (k) = V (k + qn) , 1 ≤ k ≤ qn
$ '$/"0()%"* " N$"%$/& 4.6; K$(($ /"'"7 "4)$F$ " %$(*5)&'" #%+06+#&5 '$ ($* &%)+."
8*$ σp (Hω) = ∅7 #&%& 8*&($ )"'" ω 6"/ %$(#$+)" & /$'+'& '$ =$4$(.*$; ="." )$/"(
%$(*5)&'"( 8;);# ("4%$ $:65*(?" '$ &*)"F&5"%$( #&%& "( /"'$5"( O)*%/+&0"(;
P/ !JJJ K&/&0+Q $ =$0- #*45+6&%&/ $/ G! H " ($.*+0)$ %$(*5)&'" R+6&>?" '" S%3
.*/$0)" '$ 1"%'"0 23T5"6"(UV (*#"0W& lim sup an 6= 27 "0'$ "( an (?" "( 6"$96+$0)$( '&
$:#&0(?" '$ α $/ ,%&>X$( 6"0)+0*&'&(; P0)?"7 #&%& )"'" λ $ )"'" ω ∈ Ωα7 " "#$%&'"%
R ;!U #"((*+ σp (Hω) = ∅; C"%)&0)"7 )$/"( %$(*5)&'"( *0+,"%/$(7 "* ($M&7 & &*(Y06+& '$
&*)"F&5"%$( #&%& )"'"( "( $5$/$0)"( ω '" !"" Ωα7 ("4 & 6"0'+>?" $0*06+&'& &6+/&; C"%
9/7 $/ 2ZZZ K&/&0+Q7 I+55+# $ =$0- '$/"0()%&%&/ $/ G!BH */& .$0$%&5+-&>?" '" %$(*53
)&'" "4)+'" &0)$%+"%/$0)$7 & 8*&5 "4)$F$ 8*$ #&%& )"'"( "( #&%[/$)%"( λ, α, ω7 " "#$%&'"%
Hλ,α,ω )$/ $(#$6)%" #"0)*&5 F&-+" RN$"%$/& 4.9U;
S."%& F&/"( &0&5+-&% & '$/"0()%&>?" '" N$"%$/& 4.97 & 9/ '$ "4)$% */ %$(*5)&'"
*0+,"%/$ ("4%$ & &*(Y06+& '$ &*)"F&5"%$( #&%& " "#$%&'"% '$ O6W%\'+0.$% .$%&'" #"%
#")$06+&+( O)*%/+&0"( Hλ,α,ω; C%+/$+%&/$0)$7 0")$/"( 8*$ &( #&5&F%&( sn '$90+'&( $/
R2;]U #"'$/ ($% %$5&6+"0&'&( 6"/ &( ($8*Y06+&( Vλ,α,ω '& ($.*+0)$ ,"%/&V #&%& 6&'& #&%
(n, α)7 & ($8*Y06+& Vλ,α,ω #"'$ ($% #&%)+6+"0&'& $/ #&5&F%&(7 6"/ 6&'& #&5&F%& ($0'" sn
"* sn−17 0")&0'" 8*$ #"'$/"( 6"0(+'$%&% " &5,&4$)" 0, λ7 ($0'" 0 6= λ ∈ R;
!"#$%&' ()**) #$%& n ∈ N0 = 0, 1, 2, . . . '&'() *+& (n, α)−,&-./01( '$ !+& 2!301(
f : Z→ 0, λ 45(+ 0 6= λ ∈ R 67&'(8 9 !+& :$;!<35/& '$ ,&-$: (Ij, zj), j ∈ Z= .&" ;!$>
$+ (: 5(3%!3.(: Ij = dj, dj + 1, . . . , dj+1 − 1 ⊂ Z ,&-./5/(3&+ Z?
$$+ 1 ∈ I0?
$$$+ 5&'& @"(5( zj ∈ sn, sn−1? $
$,+ & -$:.-/01( '$ f & Ij 9 zj) A:.( 9= fdjfdj+1 . . . fdj+1−1 = zj.
! !"#$%&' () !"&* !+,-.
" #$%#$&'()(' (' ('*%+#%,&-.% (%, #%/'0*&)&, 1/2$+&)0%, '+ #)3)4$), *)050&*), 6
()() #'3% 3'+) ,'72&0/'8 92' ',/: ('+%0,/$)(% '+ ;<=>?
!"# $%&'% !"! #$%$ n ∈ N0 & #$%$ ω ∈ [0, 1)' &()*#& +,! -.)/! n−0!"#)12$ (Ij, zj) %&
Vλ,α,ω3 456, %)**$' *& zj = sn−1' &.#2$ zj−1 = zj+1 = sn3 7& zj = sn' &.#2$ &()*#& +,
).#&"8!5$ I = d, d+1, ..., d+ l−1 ⊂ Z /$.#&.%$ j & %& /$,0"),&.#$ l ∈ an+1, an+1+1#!5 9+& zd = zd+1 = . . . zd+l−1 = sn & zd−1 = zd+l = sn−13
(!)*+,-. $%&/% :,! 0!5!8"! w = w1 . . . wn 6 /$.;+<!%$ %& +,! 0!5!8"! v = v1 . . . vn *&
0!"! !5<+, i ∈ 1, . . . , n' #&,$*
w1 . . . wn = vi . . . vnv1 . . . vi−1,
)*#$ 6' *& w 6 $=#)%$ %& v 0$" +,! 0&",+#!12$ />/5)/! %& *&+* *>,=$5$*3
@2'$'+%, ',/2()$ % *%+#%$/)+'0/% (), ,%32-A', () '92)-.% (' )2/%4)3%$',? B)$)
&,,%8 4)+%, )0)3&C)$ % *%+#%$/)+'0/% (' ψ ∈ l2 (Z) 92' #%(' ,'$ &04',/&7)(% )/$)46, ('
‖Ψ‖L =
(
L∑
n=1
‖Ψ(n)‖2)
12
,
L ∈ N8 ,'0(% 92'
Ψ(n) = (ψ(n+ 1), ψ(n))T ' ‖Ψ(n)‖2 = |ψ(n+ 1)|2 + |ψ(n)|2,
#%&,
1
2‖Ψ‖2L ≤ ‖ψ‖2L ≤ ‖Ψ‖2L .
D/&3&C)0(% ) 3&+&/)-.% 20&E%$+' (%, /$)-%,8 4)+%, %F/'$ ',/&+)/&4), ,%F$' % *$',G
*&+'0/% (' ‖Ψ‖L #)$) '0'$7&), 0% ',#'*/$% ' ,%32-A', 0%$+)3&C)(), H0% ,'0/&(% 92'
|ψ(0)|2 + |ψ(1)|2 = 1I () *%$$',#%0('0/' '92)-.% (' )2/%4)3%$',?
!"# $%&$% ?)(& λ, α, ω3 7+0$.@! 9+& Vλ,α,ω(j) . . . Vλ,α,ω(j + 2k − 1) 6 /$.;+<!%$ %&
(sn−1)2, (sn)
2$+ (sn−1sn)
20!"! !5<+, n ∈ N, l ≤ k' & #$%$ j ∈ 1, . . . , l3 7&;! E ∈
σ (Hλ,α,ω)3 A.#2$ #$%! *$5+12$ .$",!5)B!%! ψ %& (Hλ,α,ω − E)ψ = 0 *!#)*C!B
‖Ψ‖l+2k ≥ Dλ‖Ψ‖l
/$, Dλ =(
1 + 14C2
λ
)12, &, 9+& Cλ 6 %!%$ 0&5! "$0$*)12$ 2.193
!"! #$%&'()# *+ +%,+(-./ ,/'-$#0 !
!"#$%&'()*#+"#$%&'()( *+,-. j ∈ 1, . . . , l/ 0#) '(1$&23#4 5(.#%
Ψ(j + k) =M(λ,E, Vλ,α,ω(j) . . . Vλ,α,ω(j + k − 1))Ψ(j)
( Ψ(j + 2k) =ME(Vλ,α,ω(j) . . . Vλ,α,ω(j + 2k − 1))Ψ(j).
"#.#4 6#) 7&685(%(4 Vλ,α,ω(j) . . . Vλ,α,ω(j + 2k − 1) 9 :#$;-,*'# '( (sn−1)2, (sn)
2#-
(sn−1sn)24 5(.#%
Ψ(j + 2k) = [ME(Vλ,α,ω(j) . . . Vλ,α,ω(j + k − 1))]2Ψ(j).
<*=4 *6+&:*$'# # 5(#)(.* '( "*>+(>?@*.&+5#$4 A(.
Ψ(j + 2k)− 5)[M(λ,E, Vλ,θ,ρ(j) . . . Vλ,θ,ρ(j + k − 1))]Ψ(j + k) + Ψ(j) = 0. B /CD
E+9. '&%%#4 6(+* 0)#6#%&23# 2.194
|5)[ME(Vλ,α,ω(j) . . . Vλ,α,ω(j + k − 1))]| ≤ Cλ B /!D
6*)* *+,-. Cλ > 1/ <( B /CD ( B /!D #F5(.#%
2Cλ.*G‖Ψ(j + k)‖, ‖Ψ(j + 2k)‖ ≥ ‖Ψ(j + 2k)‖+ Cλ‖Ψ(j + k)‖≥ ‖Ψ(j)‖
6*)* 5#'# 1 ≤ j ≤ l/ H%5# &.6+&:* I-(
‖Ψ(j + k)‖2 + ‖Ψ(j + 2k)‖2 ≥ (.*G‖Ψ(j + k)‖, ‖Ψ(j + 2k)‖)2
≥ 1
4C2λ
‖Ψ(j)‖2
!"#$%&' () !"&* !+,-.
!"#" $%&% 1 ≤ j ≤ l' ())*+,
‖Ψ‖2l+2k =l+2k∑
m=1
‖Ψ(m)‖2
=l∑
m=1
‖Ψ(m)‖2 +l+2k∑
m=l+1
‖Ψ(m)‖2
≥l∑
m=1
‖Ψ(m)‖2 +l∑
m=1
(‖Ψ(m+ k)‖2 + ‖Ψ(m+ 2k)‖2)
≥l∑
m=1
‖Ψ(m)‖2 + 1
4C2λ
l∑
m=1
‖Ψ(m)‖2
=
(
1 +1
4C2λ
)
‖Ψ‖2l .
-%#$".$%, ‖Ψ‖l+2k ≥ Dλ‖Ψ‖l'
(/%#", 0)"#1+%) %) 21+") '34 1 '3 !"#" 1)$*+"# % 5#1)5*+1.$% &1 ‖Ψ‖L, 5%+ 1.1#6
/*") .% 1)!15$#% 1 )%70891) .%#+"7*:"&") ;|ψ(0)|2+|ψ(1)|2 = 1< &" 1=0"8>% &1 "0$%?"7%#1)'
!"# $%&'% !"#$ λ, α, ω #%&'(%)%'*+, E ∈ σ (Hλ,α,ω) ! ψ -$# +*.-/0* 1*%$#.'2#3# 3!
(Hλ,α,ω − E)ψ = 04 51(0*, 6#%# (*3* n ≥ 8, 7#.! # 3!+'8-#.3#3!
‖Ψ‖qn ≥ Dλ‖Ψ‖qn−8
9*$ Dλ =(
1 + 14C2
λ
)12.
(!")*+,-#./)% @"#1+%) 0)% &") *.A%#+"891) A%#.15*&") !17% 21+" '34 1 1B*C*#1+%)
=0"&#"&%) .%) !%$1.5*"*), .% )1.$*&% =01 171) )"$*)A":1+ ") D*!E$1)1) &% 21+" '3 ' -"#"
&1+%.)$#"#+%) % 71+", +%)$#"#1+%) =01
‖Ψ‖2(qn+1+qn)+qn−1 ≥ Dλ‖Ψ‖qn−4
!"#" $%&%) λ, α, ω, $%&% E ∈ σ (Hλ,α,ω), $%&") )%70891) ψ &" 1=0"8>% &1 "0$%?"7%#1), 1
$%&% n ≥ 4, !%*) qn+4 ≥ 2(qn+1 + qn) + qn−1.
@*B1 λ, α, ω 1 "7/0+ n ≥ 4' F%.)*&1#1 " n−!"#$*8>% &1 Vλ,α,ω' F%+% =01#1+%) 1B*C*#
=0"&#"&%) !"#" " %#*/1+, 5%.)*&1#1+%) %) )1/0*.$1) 5")%)G
:#+* ;' z0 = sn−1'
-17% 21+" '34, z1 = sn' F%+% sn−1 H 0+ !#1IB% &1 sn 1 z2 ∈ sn−1, sn, 1.$>% z2 = sn−1a,
!"! #$%&'()# *+ +%,+(-./ ,/'-$#0 !
"#$%& a '() *)+),-) )*-&*-.)%)/ 0&- 12/34 # *#+) 0-&*&".56& 2/789 :#(&"
z0z1z2 = sn−1snsn−1a
= sn−1sann−1sn−2sn−1a
= sn−1sann−1sn−1s
an−2−1n−3 sn−4sn−3a
= sn−1s2n−1s
an−1n−1 s
an−2−1n−3 sn−4sn−3a .
;# an ≥ 2 #$:6& z0z1z2 = sn−1s2n−1sn−1s
an−2n−1 b = sn−1s
2n−1sn−4d, <&( *)+),-)" )*-&*-.)%)"
b, d/ ;# an = 1 #$:6&
z0z1z2 = sn−1s2n−1s
an−2−1n−3 sn−4sn−3a,
# '")$%& 12/34 &=:#(&" 1#( >')+>'#- '( %&" <)"&"? an−2 = 1 &' an−2 ≥ 24 >'# z0z1z2 =
sn−1s2n−1sn−4v9 <&( '() *)+),-) )*-&*-.)%) v/ 0&-:)$:&9 )*+.<)$%& & @#() /7 <&(
l = qn−4 # k = qn−1 &=:#(&"
‖Ψ‖2(qn+1+qn)+qn−1 ≥ ‖Ψ‖qn−4+2qn−1 ≥ Dλ‖Ψ‖qn−4 .
!"# $/ z0 = sn # z1 = sn/
;# z2 = sn−1 #$:6&9 *#+& @#() /729 z3 = sn/ A*+.<)$%& ) 0-&*&".56& 2/78 &=:#(&"
z0z1z2z3 = sns2ns
an−1−1n−2 sn−3sn−29 # %# 12/34 ,#( >'#
z0z1z2z3 = sns2nsn−3w,
<&( '() *)+),-) )*-&*-.)%) w/ B)"& <&$:-C-.&9 "# z2 = sn #$:6& <&(& sn−1 D '( *-#EF&
%# sn # z3 ∈ sn−1, sn9 :#(&" z0z1z2z3 = sns2nsn−1r9 "#$%& r '() *)+),-) )*-&*-.)%)/
G)H9 *&- 12/349 z0z1z2z3 = sns2nsn−3s9 <&( '() *)+),-) s/ 0&-:)$:&9 )*+.<)$%& & @#() /7
<&( l = qn−3 # k = qn &=:#(&"
‖Ψ‖2(qn+1+qn)+qn−1 ≥ ‖Ψ‖qn−3+2qn ≥ Dλ‖Ψ‖qn−3 ≥ Dλ‖Ψ‖qn−4 .
!"# %/ z0 = sn # z1 = sn−1/
;#I)( z′j &" =+&<&" $) (n + 1)−*)-:.56& %# Vλ,α,ω/ 0#+) '$.<.%)%# %) n−*)-:.56& :#(&"
z′0 = sn+1/ B&$".%#-#(&" &" "#J'.$:#" "'=<)"&"?
!"# %&'& z′1 = sn+1.
A$)+&J)(#$:# )& <)"& 29 .":& .(*+.<) >'# s′0s′1 D "#J'.%& *&- sn+1sn−2 # %)H )*+.<)$%& &
! !"#$%&' () !"&* !+,-.
"#$% &' ()$ l = qn−2 # k = qn+1 )*+#$),
‖Ψ‖2(qn+1+qn)+qn−1 ≥ ‖Ψ‖qn−2+2qn+1 ≥ Dλ‖Ψ‖qn−2 ≥ Dλ‖Ψ‖qn−4 .
!"# $%&% z′1 = sn.
-#./# 0) "#$% &'1 2/# z′2 = sn+1& 3)4%$#5+# ()5,60#7#$), 1 ,/*(%,),&
!"# $%&%'% z′3 = sn&
3)+# 2/# #,+# (%,) )()77# ,)$#5+# ,# an+2 = 1& 8#9) "#$% &'1: z′4 = sn+1. ;)$) sn <
/$ =7#>?) 0# sn+1 # z′5 ∈ sn, sn+1: =)7 @#2A7#=#+#,+/7B +#$),
z′0z′1z′2z′3z′4z′5 = sn+1(snsn+1)
2snw′ = sn+1(snsn+1)
2sn−1san−1n−1 sn−2w
′
()$ /$% =%9%47% %=7)=76%0% w′& C=96(%50) ) "#$% &' ()$ l = qn−1 # k = qn + qn+1
)*+#$),
‖Ψ‖qn−1+2(qn+qn+1) ≥ Dλ‖Ψ‖qn−1 ≥ Dλ‖Ψ‖qn−4 .
!"# $%&%&% z′3 = sn+1&
;)5,60#7# %, ()5,#2/D5(6%, 0#,+# (%,) =%7+6(/9%7 =%7% ), *9)(), 5% n−=%7+6EF)& G#$),
z0z1 . . . z2an+1+4 = snsn−1snsan+1n sn−1s
an+1n sn−1.
;)$) sn < /$ =7#>?) 0# sn+1: #,+# *9)() 0#4# ,#7 ,#./60) =)7 sn& 8)7+%5+): +#$), %
,#2/D5(6% 0# *9)(),
snsn−1snsan+1n sn−1s
an+1n sn−1sn. @ & B
H,%50) % 87)=),6EF) 1&'I =)0#$), 7##,(7#4#7 @ & B ()$)
snsn−1snsan+1n sn−1s
an+1n sns
an−1−1n−2 sn−3sn−2,
) 2/%9 =)0# ,#7 65+#7=7#+%0) ()$)
snsn−1snsan+1n sn−1sns
an+1n s
an−1−1n−2 sn−3sn−2.
C.)7%: )*,#74# 2/# sn−1snsan+1n < ()5J/.%0) 0# snsn+1 = sns
an+1n sn−1& C,,6$: %=96(%50) )
"#$% &' ()$ l = qn−3 # k = qn + qn+1 )*+#$),
‖Ψ‖2(qn+1+qn)+qn−1 ≥ ‖Ψ‖qn−3+2(qn+qn+1) ≥ Dλ‖Ψ‖qn−3 ≥ Dλ‖Ψ‖qn−4 .
;)$) ), (%,), ': 1 # K ()*7#$ +)0%, %, =),,L4#6, #,()9M%, 0# z0, z1: ) 9#$% #,+N 0#$)5,O
!"! #$%&'()# *+ +%,+(-./ ,/'-$#0 !
"#$%&'
() *$#"+,-.$#/ & 01)$ '23 *1#)+"1 14,.-+# &5 $-"&6$. %& 15*1,"#& %1 Hλ,α,ω/ *$#$
"&%&5 &5 *$#7)1"#&5 λ, α, ω. 8$+5 *#1,+5$)19"1/ "1)&5
!"#$%&'()*#+:,!#'!"( 4.9; <1=$) λ, α, ω $#>+"#?#+&5/ E ∈ σ (Hλ,α,ω) 1 ψ -)$ 5&.-@A&
9&#)$.+B$%$ %1 (Hλ,α,ω − E)ψ = 0' (9"A& *1.& 01)$ '23/ "1)&5
‖Ψ‖q8n ≥ Dλ‖Ψ‖q8n−8 ≥ . . . ≥ Dnλ‖Ψ‖q0 = Dn
λ‖Ψ‖1 = Dnλ , ∀n ≥ 1.
C5"& +)*.+,$ D-1
‖Ψ‖2l2 ≥ ‖Ψ‖2q8n ≥ D2nλ , ∀n ≥ 1, ,&) Dλ > 1.
E$B19%& n→∞ &>"1)&5
‖Ψ‖2l2 =∞∑
m=1
‖Ψ(m)‖2 =∞.
F55+)/ *$#$ "&%&5 &5 *$#7)1"#&5 λ, α, ω/ 9A& 14+5"1 5&.-@A& ψ 1) l2' G&#"$9"&/ *$#$ "&%&5
&5 *$#7)1"#&5 λ, α, ω/ & &*1#$%&# Hλ,α,ω "1) 15*1,"#& *&9"-$. 6$B+&'
-#.!/# 01/23()*# .! 4!'5#.#6 H15-."$%&5 5&>#1 $ $-5I9,+$ -9+J&#)1 %1 $-"&6$K
. "$)>L) J&#$) &>"+%&5 *$#$ $ ,.$551 %1 *&"19,+$+5 %-*.+,$@A& %1 *1#M&%& 1) N22O/ &
D-$. -"+.+B$ &5 $#P-)19"&5 %1 Q&#%&9 *$#$ &>"1# 15"1 #15-."$%&'
R$)&5 $9$.+5$# & )&%1.& %1 <,S#T%+9P1# Hω %$ J&#)$ :U'U; P1#$%& *&# *&"19,+$+5
%-*.+,$@A& %1 *1#M&%&/ %1V9+%& 9$ 51@A& U'U' W&95+%1#$9%& $ 5->5"+"-+@A& *#+)+"+6$ %-K
*.+,$@A& %1 *1#M&%&
ζdp (a) = ab ζdp (b) = aa,
5&>#1 & $.J$>1"& A = a, b/ %&9%1 "1)&5 $5 51P-+9"15 14"195X15 9$"-#$+5 *&# ,&9,$"19$@A&
ζndp (a) = ζn−1dp (ab) = ζn−1dp (a) ζn−1dp (b) ,
ζndp (b) = ζn−1dp (aa) = ζn−1dp (a) ζn−1dp (a) .
R$)&5 %1V9+# sn = ζndp (a) 1 tn = ζndp (b)/ %&9%1 "1)&5 $5 51P-+9"15 #1.$@X15
sn = sn−1tn−1 1 tn = sn−1sn−1. : '3;
R1#+V,$K51 J$,+.)19"1 *&# +9%-@A& 5&>#1 n/ D-1 $5 *$.$6#$5 sn 1 tn *&55-1) "$)$9S& 2n'
Y$)>L) "1)&5 *1.$ *#&*&5+@A& $>$+4& D-1 155$5 *$.$6#$5 5A& D-$51 +%I9"+,$5 NZO'
! !"#$%&' () !"&* !+,-.
!"#"$%&'" ()*+) !"! #$%$ n ∈ N !& '!(!)"!& sn * tn &+$ !& ,*&,!&- *./*#$ '!"! &0!
"*&'*/#1)! 2(#1,! (*#"! ! %1"*1#!3
,-."/$0!1&'")"#$%&'() * *+,&-*./( h : Al → Al−1+(0 h (a1 . . . al) = a1 . . . al−11 +*0*
l > 12
3%4/(1 +*0* 5#'(%)40*0 #))* +0(+()&./( 6*)4* 7#0&$-*0 89# h(
ζndp (a))
= h(
ζndp (b))
2
:(' #;#&4(1 9)*%5( ( +0&%-<+&( 5# &%59./( )(60# n1 4#'() 89# +*0* n = 1 ( 0#)9,4*5( =
&'#5&*4(2 >?(0*1 +#,* 5#$%&./( 5* *+,&-*./( h # +#,* @&+A4#)# 5# &%59./( 4#'() 89#
h(
ζndp (a))
= h(
ζn−1dp (a) ζn−1dp (b))
= ζn−1dp (a)h(
ζn−1dp (a))
= h(
ζn−1dp (a) ζn−1dp (a))
= h(
ζndp (b))
.
B#'() ( )#?9&%4# 0#)9,4*5( )(60# () 40*.() 5*) '*40&C#) 5# 40*%);#0D%-&* -(%)409<5*)
+#,( +(4#%-&*, ?#0*5( +#,* )96)4&49&./( 59+,&-*./( 5# +#0<(5(2
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,
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(
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1 0
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[
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npk′
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)]
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++@ V2 5 03# :0,;<$ !4(#'#B
+++@ V2 (x) = |x|−1/29 %#&# −1/2 ≤ x ≤ 1/29
4!,'$ α, θ ∈ [0, 1) ! α 03 ,C3!&$ '! D+$08+66!E F,.<$9 $ !4%!(.&$ %$,.0#6 '! H 5 8#>+$E
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D, δ > 0 >"+, &'%
|f (z)− f (y)| ≤ D |z − y|δ ,
!"#" &'"+,&'%# z, y ∈ RA O+H")-/ / %,*"1"# C8 >%:/,
limm→∞
eCqm∫ 2qm
−qm|V2 (xα + θ)− V2 (xαm + θ)| dx ≤
≤ limm→∞
eCqmD |α− αm|δ∫ 2qm
−qm|x|δ dx
≤ limm→∞
eCqmD|qm|δ+1 + |2qm|δ+1
δ + 1m−qm = 0.
3%,,% :/-/8 ,% ", $')5;%, V2 ,6/ DE1-%# */)>C)'", %)>6/ !%1/ P/#/1Q#+/ 3.8 / /!%#"<
-/# -% R*M#E-+)@%# -/ >+!/ 2SATT7 @%#"-/, !/# !/>%)*+"+, -" $/#:" 2SATU7 )6/ !/,,'+
"'>/0"1/#%,A V,,+:8 >%:/, 0%#+4*"-/ " V!1+*"56/ TAUA
! !"#$%&' () !"&* !+,-.
""# $%&'"()*+&(% ,-) V2 . -/+ 0-&12% )'3+(+4 5)/%' ,-) )6"'5) -/+ 7+*5"12% −qm =
x0 < x1 < · · · < xn = 2qm ')&(% 7%''89): )'3*)9)*
V2 (x) =n
∑
i=1
aiχAi(x) ,
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(+ *)5+< $%/% α . -/ &=/)*% () >"%-9"::)4 5)/%' ,-) 7+*+ 5%(% x ∈ R
|αx− αmx| ≤1
mqm|x| m→∞−→ 0.
?)'') /%(%4 3%&'"()*+&(% -/ m '-@3")&5)/)&5) A*+&() () /+&)"*+ ,-) ') αx ∈ Ai5)/%' ,-) αmx ∈ Ai4 )&52%
|V2 (αx)− V2 (αmx)| =∣
∣
∣
∣
∣
n∑
i=1
ai (χAi(αx)− χAi
(αmx))
∣
∣
∣
∣
∣
= 0.
>%A% + )67*)''2% BC<D# ')A-) "/)("+5+/)&5)< E%*5+&5%4 3%&3:-8/%' ,-) ') %' 7%5)&F
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"""# O"6+&(% % )'3+:+* C4 5)/%'
limm→∞
eCqm∫ 2qm
−qm|V2 (xα + θ)− V2 (xαm + θ)| dx =
= limm→∞
eCqm
∣
∣
√α−√αm
∣
∣
√α√αm
∫ 2qm
−qm
1√
|x|dx
= limm→∞
eCqm(
2 + 2√2)√
qm |αm − α|√αm√α(√
αm +√α)
≤ limm→∞
eCqm(
2 + 2√2)√
qm√αm√α(√
αm +√α)m−qm = 0.
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0-&12% V2 () 7)*8%(% L4 ()@&"(+ 7%* V2 (x) = |x|−γ4 7+*+ −1/2 ≤ x ≤ 1/24 ')&(% 0 < γ < 1
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