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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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(

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)

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$67)5$& 8,)

ME,ω (n) =

(

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→∞

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→∞

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(

(

ζnf (a)))

yn = tr(

(

ζnf (b)))

zn = tr(

(

ζnf (a))

(

ζnf (b)))

7,0!$ 1&,

xn = tr(

(

ζn−1f (a))

(

ζn−1f (b)))

= zn−1

yn = tr(

(

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(

(

ζn−1f (a))

(

ζn−1f (b))

(

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(

(

ζndp(a)))

yn = tr(

(

ζndp(b)))

7$#6%! ! 8.!/.2# %. 9#-.:;<#2)-(!61 !'(.2!$ 3&.

yn = tr(

(

ζn−1dp (a))2)

= tr(

(

ζn−1dp (a)))

tr(

(

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(

(

ζn−1f (a))

(

ζn−1f (b)))

= tr(

(

ζn−1f (a)))

tr(

(

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+,*+ (+*+$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@ '

!"! #$%&'$& ()&*%$+,& !

"#$# %#&# "'()*%+#, ,+-+(#&' V : Z→ R #..'%+#-'. ' '")$#&'$ &) /%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

I*(:'

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)‖ ≥

)* 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

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 #*=(* .(*

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)‖ ≥

>@()#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)‖ ≥

8().#,.(2 ;#)# .(-( nk2

max (‖Ψ(2nk)‖ , ‖Ψ(nk)‖ , ‖Ψ(−nk)‖) ≥1

>&&1$2 *+#,-( nk −→ ∞ .%$(& *+% ‖Ψ(nk)‖ −→ 02 ;(1& ψ ∈ l2 (Z)6 C($( ,( .%()%$#

#,.%)1() 1&.( ,(& 9%B# # +$# =(,.)#-1/0(2 ;(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$%), &,,$50&4$, &$ $;)%&4$%

!"#!$% &'(' !"#$ 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

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

*!$"+,-#%./"'O)C$ 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

! !"#$%&' () !*+%,-.$'/ 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

!"#$%&'()*#+,-!#'!"( 3.2. =)>&9 ψ ) ψm ($+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

!" #$%&'( )(*+,*-.

A.<2$ <.=;' B$4'< .<2#&$+ #4$ *.%.+$/0C$=;' &' +.<#/2$&' &. D'+&'% '+0*0%$/ -$+$ '

'-.+$&'+ &. E1F+G&0%*.+ &$ 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+'

2.4 2.%&O%10$ $ <.+ <0%*#/$+ 1'%2@%#'( K$+201#/$+4.%2., .<2$4'< 0%2.+.<<$&'< .4 4L2'M

&'< "#. -.+402.4 .P1/#0+ '< $#2'B$/'+.< $<<'10$&'< $ H, '# <.Q$, 'I2.+4'< "#. ' .<-.12+'

-'%2#$/ &. H L B$C0'(

D'+&'% R 6S &.4'%<2+'# "#. ' '-.+$&'+ H %;' -'<<#0 $#2'B$/'+.< <. V L #4 -'2.%10$/

&. D'+&'% 5T.U%0=;' 1.19( K$+$ ' 1$<' &0<1+.2' .<2#&$4'< %$< <.=>.< $%2.+0'+.< $ .<2.

1$-@2#/' $/*#%< 4L2'&'< "#. -.+402.4 .P1/#0+ '< $#2'B$/'+.< $<<'10$&'< $' '-.+$&'+ &.

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

|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

(

ψ′1 (t)− ψ′2 (t)

(W1 (t)− E)ψ1 (t)− (W2 (t)− E)ψ2 (t)

)

dt =

∫ x

(

(W1 (t)−W2 (t))ψ2 (t)

)

dt+

∫ x

(

ψ′1 (t)− ψ′2 (t)

(W1 (t)− E) (ψ1 (t)− ψ2 (t))

)

dt =

∫ x

(

(W1 (t)−W2 (t))ψ2 (t)

)

dt+

∫ x

(

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

|W1 (t)−W2 (t)| |ψ2 (t)| dt

∫ x

∥

(

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 1

W1 (t)− E 0

)∥

∥

dt∫ x

|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

'!"()*+,#-.(%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

!"! #$%&'( )(*+,*-. !

"# |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

!"#$%&'()* ?/&/ (/ 0'+/ %)+0*#./$ +# 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)

)

(

)

(

ϕ (x)

ϕ′ (x)

)

(

)

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

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

/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,#;

!"#$%&'

!"#$%&'()

!"#! $%&'#()* +%,*" !"#(-%. %)/(,%" %&)0$%12!" -*" %./(,!3#*" -! 4*.-*3 %3%)0"%-*"

3* $%&'#()* %3#!.0*.5 6)7, -0""*8 !"#(-%.!,*" &.*&.0!-%-!" #%0" $*,* 9!"&!$#.* $*,

,!-0-% -! :!;!"/(! <!.*=8 "!3-* >(! !""% &.*&.0!-%-! ?(3#%,!3#! $*, % %("@3$0% -!

%(#*+%)*.!"8 &%.% (, *&!.%-*. -! A$B.C-03/!. H8 &!.,0#!, $*3$)(0. >(! * !"&!$#.* -!"#!

*&!.%-*. 7 &(.%,!3#! "03/()%. $*3#'3(*5

6" %&)0$%12!" -*" %./(,!3#*" -! 4*.-*3 DE;)*$*" ! FE;)*$*" G H!*.!,%" I5D ! I5FJ

"K* &%.% (,% L%,')0% -! *&!.%-*.!" -! A$B.C-03/!. IM $*, &*#!3$0%0" /!.%-*" &*. "(;"E

#0#(012!" &.0,0#0+%" ! &*. .*#%12!" 3% $0.$(3L!.@3$0% S18 "!3-* >(! *" .!"()#%-*" "*;.!

% %("@3$0% -! !"&!$#.* &*3#(%) &%.% !""!" ,*-!)*" "K* *;#0-*" "*; #.@" &*3#*" -! +0"#%

-0L!.!3#!"N .!"()#%-*" /!37.0$*"8 !"!# ! (30L*.,!"8 -*3-! +%,*" %3%)0"%. $*, (, ,%0*.

-!#%)B%,!3#* *" .!"()#%-*" (30L*.,!" &%.% *" ,*-!)*" /!.%-*" &*. &*#!3$0%0" A#(.,0%3*"

! &!)% "(;"#0#(01K* &.0,0#0+% -(&)0$%1K* -! &!.'*-*5 O%.% % +!."K* -! 4*.-*3 !"#(-%-% 3*

H!*.!,% 3.2 #!,*" >(! * ,*-!)* -! A$B.C-03/!. 6),*"#EP%#B0!(8 &%.% $!.#%" .!%)0<%12!"

-* &*#!3$0%)8 &*""(0 !"&!$#.* &*3#(%) +%<0*5

Q" ,*-!)*" -! A$B.C-03/!. IM $*3#'3(* %""*$0%-*" $*, &*#!3$0%0" -! 4*.-*3 /!3!.%E

)0<%-*" 3K* &*""(!, %(#*+%)*.!" GH!*.!,% 1.4J8 "!3-* %" %&)0$%12!" -!""! .!"()#%-* &%.%

&*#!3$0%0" >(%"!E&!.0R-0$*" /!.%-*" &*. L.!>(@3$0%" -! :0*(+0))!8 *" ,*-!)*" *;#0-*" &*.

L(312!" SC)-!. $*3#'3(%"8 L(312!" -* #0&* !"$%-% ! L(312!" $*, "03/()%.0-%-!" )*$%0"5

!"! #$%&'()# *+ +%,+(-./ ,/'-$#0 !

!" #$%&'()*+% %+,-$ .&%/012) *$ 3%4$1(-+ 5+0(&)'

"#$%& $#'() *&+)$ #$%,-&. )$ .#$,/%&-)$ $)0.# & &,$1234& -# #$5#3%.) 5)2%,&/ 5&.& )$

+)-#/)$ 6#.&-)$ 5). $,0$%4%,4'7#$ 5.4+4%4*&$ # .)%&'7#$ 2& 34.3,28#.1234&9 :.&%43&+#2%#

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E? lim supn→∞G (n) ⊆ Ωc ,

F? lim supµ (G (n)) > 0.

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#")$06+&+( O)*%/+&0"( Hλ,α,ω; C%+/$+%&/$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)

L ∈ N8 ,'0(% 92'

Ψ(n) = (ψ(n+ 1), ψ(n))T ' ‖Ψ(n)‖2 = |ψ(n+ 1)|2 + |ψ(n)|2,

#%&,

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

)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

)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 %& 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

.&#15 "$)>L) J&#$) &>"+%&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))

:(' #;#&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#

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

!"#"$%&'" ()*2) !"! #$%$ E ∈ σ (Hω) * #$%$ n ∈ N- #*,$&

min |xn| , |xn+1| ≤ 2.

:(%)&5#0#'() ( ;*4( 89# +*0* 4(5( n1 4(5( ω ∈ Ω +(5# )#0 9%&-*'#%4# 5#-('+()4( #'

+0(594( &%$%&4( 5# 6,(-() 5* ;(0'* sn (9 tn2 :@*'*'() #)4* 5#-('+()&./( 5# nE+*04&./(

5# ω2 F#)9'&'() *) +0(+0&#5*5#) %#-#))G0&*) %* +0(+()&./( * )#?9&01 -9H* 5#'(%)40*./(

#%-(%40*E)# #' IJJK2

!"#"$%&'" ()*3) !"! #$%$ n- #$%$ ω ∈ Ω #*, 0,! 241/! n5'!"#16+$3 7$ '"$%0#$ %*&#!

"*'"*&*4#!6+$- 0, tn58($/$ 9 &*,'"* 1&$(!%$ * *4#"* %$1& /$4&*/0#1)$& tn58($/$& *.1&#* 0,

$0 #":& sn58($/$&3

4-"!-.1 ()*5) ; $'*"!%$" %* </=">%14?*" Hω %! @$",! LM2MN- ?*"!%$ '$" '$#*4/1!1&

%0'(1/!6+$ %* '*"A$%$- '$&&01 *&'*/#"$ '$4#0!( )!B1$- '!"! #$%$ ω ∈ Ω3

,-."/$0!1&'") O&P# ω ∈ Ω1 E ∈ σ (Hω)1 # 9'* )(,9./( ψ 5#

(Hω − E)ψ = 0

)*4&);*C#%5( |ψ (0)|2 + |ψ (1)|2 = 12

Q*'() '()40*0 89# E %/( +(5# )#0 9' *94(7*,(0 +*0* ( (+#0*5(0 Hω1 (9 )#H*1 7#0&$-*0

89# ψ %/( +#04#%-# *( #)+*.( l2 (Z)1 # +*0* &))( 6*)4* 7#0&$-*0 89# +*0* ( (+#0*5(0 Hω

!"! #$%&'()# *+ +%,+(-./ ,/'-$#0 !

"#$%&' ('$ (')#*+,%,- &.(/,+%01' &# (#$2'&' '- %$".3#*)'- &# 4'$&'* 567/'+'- '. 867/'+'-

9:#'$#3%- ;<5 '. ;<8= -1' -%),->#,)'- #3 +%&% +%-'<

?'3 #>#,)'@ -#A% k ∈ N # n ∈ N )%/ B.# 2n ≥ k@ # +'*-,&#$#3'- % n6(%$),01' &# ω<

!"# $% snsn<

C%3'- +'*-,&#$%$ ' +%-' &# .3 sn67/'+' -#".,&' % &,$#,)% ('$ .3 '.)$' sn67/'+'@ -#*&'

B.# ,*&,+%$#3'- +'3' sn ' 7/'+' '*&# #-)D /'+%/,E%&% ('*)' 0 ∈ Z< F),/,E%*&' % G$'('6

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eCqm

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|x|dx

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αm +√α)

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