Spectral analysis of the Kronig-Penney model with Wigner-von Neumann perturbations

0.26 MB·English

by  Vladimir Lotoreichik

SPECTRAL ANALYSIS OF THE KRONIG-PENNEY MODEL
WITH WIGNER-VON NEUMANN PERTURBATIONS
VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
Abstract. Thespectrum ofthe self-adjointSchro¨dinger operator associated
with the Kronig-Penney model on the half-linehas a band-gap structure: its
2 absolutelycontinuousspectrumconsistsofintervals(bands)separatedbygaps.
1 Weshowthatifonechangesstrengthsofinteractionsorlocationsofinteraction
0 centersbyaddinganoscillatingandslowlydecayingsequencewhichresembles
2 the classical Wigner-vonNeumann potential, then this structure of the abso-
n lutely continuous spectrum is preserved. At the same time in each spectral
a bandpreciselytwocriticalpointsappear. Atthesepoints“instable”embedded
J eigenvalues may exist. We obtain locations of the critical points and discuss
foreachofthemthepossibilityofanembeddedeigenvaluetoappear. Wealso
9
showthatthespectrumingapsremainspurepoint.
1

]
P
S
. 1. Introduction
h
t
a In the classical paper [vNW29] von Neumann and Wigner studied the one-
m dimensional Schr¨odinger operator with the potential of the form csin(2ωx) and dis-
x
[ coveredthatsuchanoperatormayhaveaneigenvalueatthepointofthecontinuous

spectrum λ=ω2. Since then such potentials permanently attracted the interest of
2
different authors [Al72, Ma73, RS78, Be91, Be94, HKS91, KN07, N07, L10, JS10].
v
7 Potentials of the type csin(2ωx) with γ (0,1) also possess such a property and
1 they are often called Wignxeγr-von Neuma∈nn potentials.
47 Consider the operator Hκ on R+ that formally corresponds to the differential
expression
.
2
d2
1
+ α δ δ ,
1 −dx2 0 ndh nd ·i
n∈N
1 X
: with the boundary condition
v
Xi (1.1) ψ(0)cosκ ψ′(0)sinκ =0,
−
ar at the origin, where α0 R,d > 0,κ [0,π) and δx is the delta-distribution
supported on the point x∈ R (for the m∈athematically rigorous definition of the
+
∈
operator see Section 2.1, cf. [AGHH05, Chapter III]). The operator Hκ is self-
adjoint in L2(R ). It describes the behavior of a free non-relativistic charged
+
quantum particle interacting with the lattice dN. The constant α characterizes
0
the strength of interaction between the free charged quantum particle and each
interaction center in the lattice. The spectrum of the operator Hκ has a band-gap
structure: it consists of infinitely many bands of the purely absolutely continuous
spectrumandoutsidethesebandsthespectrumofHκ isdiscrete. Theinvestigation
ofsuchoperatorswasinitiatedintheclassicalpaper[KP31]byKronigandPenney.
In the present paper we study what happens with the spectrum of the Kronig-
Penney model in the case of perturbation of strengths or positions of interactions
by a slowly decaying oscillating sequence resembling the Wigner-von Neumann
potential. Let constants d,α0,c,ω,γ and a real-valued sequence qn n∈N be such
{ }
1
2 VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
that
π 1
(1.2) d>0,α0 R,ω 0, ,γ ,1 , qn n∈N ℓ1(N).
∈ ∈ 2 ∈ 2 { } ∈
(cid:16) (cid:17) (cid:18) (cid:21)
ModelI:Wigner-vonNeumannamplitudeperturbation. WeaddadiscreteWigner-
von Neumann potential to the constant sequence of interaction strengths. Let the
discrete set X = x : n N supporting the interaction centers be defined as
n
{ ∈ }
(1.3) x :=nd, n N,
n
e e ∈
and let the sequence of interaction strengths α= αn n∈N be as below:
e { }
csin(2ωn)
(1.4) α :=α + +q , n N.
n 0 nγ e n e ∈
We study the self-adjoient operator Hκ,Xe,αe which formally corresponds to the dif-
ferential expression
d2
−dx2 + αnδxenhδxen,·i
n∈N
X
withtheboundarycondition(1.1). Thisopeeratorreflectsanamplitudeperturbation
of the Kronig-Penney model.
ModelII:Wigner-vonNeumannpositionalperturbation. Wechangethedistances
betweeninteractioncentersina“Wigner-vonNeumann”way,i.e. weaddasequence
of the form of Wigner-von Neumann potential to the coordinates of interaction
centers leaving the strengths constant. Let the discrete set X = x : n N be as
n
{ ∈ }
below:
(1.5) x :=nd+ csin(2ωn) +q , n N,b b
n nγ n ∈
and the sequence of strengths be defined as
b
(1.6) α :=α , n N.
n 0
∈
We study the operator Hκ,Xb,αb which formally corresponds to the differential ex-
pression b
d2
−dx2 + αnδxbnhδxbn,·i
n∈N
X
withthe boundarycondition(1.1). Thisobperatorreflectsapositionalperturbation
of the Kronig-Penney model and describes properties of one-dimensional crystals
with global defects. We also mention that local defects in the Kronig-Penney are
discussed in [AGHH05, III.2.6]; situations of random perturbations of positions
§
were recently considered in [HIT10].
The essential spectrum of the operator Hκ has a band-gap structure similar to
the case of Schr¨odinger operator with regular periodic potential:
∞
σess(Hκ)=σac(Hκ)= λ+2n−1,λ−2n−1 ∪ λ−2n,λ+2n ,
n[=1(cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
where λ+ <λ− λ− <λ+ λ+ <λ− λ− <λ+ ...
1 1 ≤ 2 2 ≤···≤ 2n−1 2n−1 ≤ 2n 2n ≤
The locations of boundary points of the spectral bands are determined by the
parametersα andd. Namely,thevaluesλ± arethen-throotsofthecorresponding
0 n
Kronig-Penney equations
L √λ = 1,
δ
±
where (cid:16) (cid:17)
sin(kd)
(1.7) L (k)=cos(kd)+α .
δ 0
2k
WIGNER VON-NEUMANN PERTURBATION OF THE KRONIG-PENNEY MODEL 3
L (k)
δ
L (k)=α sin(kd) +cos(kd)
δ 0 2k
1
cos(ω)
x x x x x x x k
cos(ω)
−
1
−
Figure 1. The curve is the graph of L ; bold intervals are bands
δ
of the absolutely continuous spectrum; crosses denote the critical
points {λ±n,cr}n∈N.
For the details the reader is referred to the monograph [AGHH05, Chapter III].
In the present paper we show that the absolutely continuous spectra of the
operatorsHκ,Xe,αe andHκ,Xb,αb coincidewiththeabsolutelycontinuousspectraofthe
non-perturbedoperatorHκ. Howeverthespectruminbandsmaynotremainpurely
absolutely continuous. Namely, at certain (critical) points embedded eigenvalues
may appear. In each band there are two such points. The critical points λ± in
n,cr
the n-th spectral band are the n-th roots of the equations
L √λ = cosω.
δ
±
(cid:16) (cid:17)
The illustration is given on the Figure I.
For the operatorsconsideredwe give exact conditions which ensure that a given
critical point is indeed an embedded eigenvalue for some κ [0,π). This can
occur only for one value of κ [0,π). The possibility of the∈appearance of an
∈
embeddedeigenvalueatcertaincriticalpointdependsontherateofthedecayofthe
subordinate generalized eigenvector. We calculate the asymptotics of generalized
eigenvectorsfor all values of the spectral parameter λ including the critical points,
excepttheendpointsofthebands. Wealsoshowthatthespectrumingapsremains
pure point.
Ourresultsareclosetotheresultsforone-dimensionalSchr¨odingeroperatorwith
the Wigner-von Neumann potential and a periodic background potential. Such
operators were considered very recently in [KN07, NS11, KS11].
Tostudyspectrainbandswemakeadiscretizationofthespectralequationsand
furtherweperformanasymptoticintegrationofobtaineddiscretelinearsystemus-
ing Benzaid-Lutz-type theorems [BL87]. As the next step we apply a modification
of Gilbert-Pearson subordinacy theory [GP87]. To study spectra in gaps we ap-
ply compact perturbation argument in Weyl function method, see, e.g., [DM95],
[BGP08].
The reader can trace some analogies of our case with Jacobi matrices. The
coefficient matrix of the discrete linear system that appears in our analysis has a
form similar to the transfer-matrix for some Jacobi matrix and the Weyl function
that appears in our analysis takes Jacobi matrices as its values.
4 VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
Thebodyofthepapercontainstwoparts: thepreliminarypart—whichconsists
of mostly known material — and the main part, where we obtain new results. In
the preliminary part we give a rigorous definition of one-dimensional Schr¨odinger
operators with δ-interactions (Section 2.1), show how to reduce the spectral equa-
tions for these operators to discrete linear systems in C2 (Section 2.2), provide a
formulation of the subordinacy theory analogue for the operators considered (Sec-
tion 2.3). Further we formulate few results from asymptotic integration theory for
discrete linear systems (Section 2.4) and providea characterizationof the essential
spectra of considered operators using the Weyl function associated with certain
ordinary boundary triple (Section 2.5). In the main part, in Section 3.1 we study
a special class of discrete linear systems in C2 and find asymptotics of solutions
of these systems. After certain technical preliminary calculations in Section 3.2,
weproceedtoSection3.3, whereweobtainasymptoticsofgeneralizedeigenvectors
for Schr¨odinger operatorswith point interactions subject to Model I and Model II.
Furtherwepasstothe conclusionsaboutthe spectrainbandsputting anemphasis
oncriticalpoints. Section3.4isdevotedtotheanalysisofspectraingapsbymeans
of the Weyl function.
Notations. By small letters with integer subindices, e.g. ξ , we denote sequences
n
of complex numbers. By small letters with integer subindices and arrows above,
e.g. −→un,wedenotesequencesofC2-vectors. Bycapitalletterswithintegerindices,
e.g. R , we denote sequences of 2 2 matrices with complex entries. We use
n
notationsℓp(N), ℓp(N,C2)andℓp(N,C×2×2)forspacesofsummable (p=1),square-
summable (p = 2) and bounded (p = ) sequences of complex values, complex
∞
two-dimensional vectors and complex 2 2 matrices, respectively. We denote for
×
a self-adjoint operator its pure point, absolutely continuous, singular continuous,
essential and discrete spectra by σ , σ , σ , σ and σ , respectively. We write
pp ac sc ess d
T S in the case T is a compact operator. Other notations are assumed to be
∞
∈
clear for the reader.
2. Preliminaries
2.1. Definition of operators with point interactions. In this section we give
a rigorous definition of operators with δ-interactions, see, e.g., [GK85, Ko89]. Let
α= αn n∈N be asequenceofrealnumbersandletX = xn: n N be adiscrete
set o{n R} ordered as 0<x <x <.... Assume that th{e set X∈sat}isfies
+ 1 2
(2.1) inf x x >0 and sup x x < .
n+1 n n+1 n
n∈N − n∈N − ∞
(cid:12) (cid:12) (cid:12) (cid:12)
Denotealsox0 :=0.(cid:12)Inorderto(cid:12)definetheoperato(cid:12)rcorrespon(cid:12)dingtotheexpression
d2
τ := + α δ δ ,
X,α −dx2 n xnh xn ·i
n∈N
X
and the boundary condition
ψ(0)cosκ ψ′(0)sinκ =0,
−
consider the following set of functions:
:= ψ : ψ,ψ′ AC (R X), ψ(xn+)=ψ(xn−)=ψ(xn) , n N
SX,α ( ∈ loc +\ ψ′(xn+)−ψ′(xn−)=αnψ(xn) ∈ )
and let the operator Hκ,X,α be defined by its action
Hκ,X,αψ := ψ′′
−
on the domain
domHκ,X,α := ψ L2(R+) X,α: ψ′′ L2(R+),ψ(0)cosκ =ψ′(0)sinκ .
∈ ∩S ∈
(cid:8) (cid:9)
WIGNER VON-NEUMANN PERTURBATION OF THE KRONIG-PENNEY MODEL 5
The spectral equation τ ψ =λψ is understood as the equation ψ′′(x)=λψ(x)
X,α
−
for ψ . The latter is equivalent to the following system:
X,α
∈S
ψ′′(x)=λψ(x), x R X,
+
(2.2) − ∈ \
ψ(x +)=ψ(x ), ψ′(x +)=ψ′(x )+α ψ(x ), n N.
n n n n n n
− − − ∈
Theequation(2.2)hastwolinearindependentsolutionswhicharecalledgeneralized
eigenvectors. If ψ L2(R ) satisfies (2.2) and the boundary condition, then ψ is
+
∈
an eigenfunction of Hκ,X,α.
2.2. Reduction of the eigenfunction equation to a discrete linear system.
In this subsection we recall rather well-known way of reduction of the spectral
equation (2.2) to a discrete linear system. Let the discrete set X and the sequence
of strengths α be as in the previous section. To make our formulas more compact
we introduce the following notations
s (k):=sin k(x x ) and c (k):=cos k(x x ) , n N .
n n+1 n n n+1 n 0
− − ∈
For a solution ψ(cid:0) of (2.2) corre(cid:1)spondingto λ=k2 w(cid:0)e introduce th(cid:1)e sequence below
ξ :=ψ(x ), n N .
n n 0
∈
If the condition (2.1) is satisfied, then by [AGHH05, Chapter III, Theorem 2.1.5]
for k C R and n N such that s (k),s (k)=0 it holds that
+ n−1 n
∈ ∪ ∈ 6
ξ ξ c (k) c (k)
n−1 n+1 n−1 n
(2.3) k + + α +k + ξ =0.
n n
− sn−1(k) sn(k)! sn−1(k) sn(k) !
(cid:16) (cid:17)
Inversely,solutionsoftheeigenfunctionequationoneachoftheintervals(x ,x )
n n+1
can be recoveredfrom their values at the endpoints x and x :
n n+1
ξ sin(k(x x))+ξ sin(k(x x ))
n n+1 n+1 n
(2.4) ψ(x)= − − , x [x ;x ].
n n+1
sin(k(x x )) ∈
n+1 n
−
The reader may confer with [E97], where a more general case of a quantum graph
is considered.
Insteadofworkingwithrecurrencerelation(2.3)wewillconsideradiscretelinear
system in C2. Define
ξ
−→un := nξ−1 .
n
(cid:18) (cid:19)
Then (2.3) is equivalent to
(2.5) −→un+1 =Tn(k)−→un
with
0 1
(2.6) Tn(k):= sn(k) sin(k(xn+1−xn−1)) + αnsn(k)
−sn−1(k) sn−1(k) k !
The coefficient matrix of this system T (k) is called the transfer-matrix.
n
2.3. Subordinacy. The subordinacy theory as suggested in [GP87] by D. Gilbert
andD.Pearsonproducedastronginfluenceonthespectraltheoryofone-dimensional
Schr¨odinger operators. Later on the subordinacy theory was translated to differ-
enceequations[KP92]. ForSchr¨odingeroperatorswithδ-interactionsthereexistsa
modificationofthesubordinacytheory,see,e.g.,[SCS94]whichrelatesthespectral
properties of the operator Hκ,X,α with the asymptotic behavior of the solutions of
the spectral equation (2.2). Analogously, to the classical definition of the subordi-
nacy [GP87] we say that a solution ψ of the equation τ ψ =λψ is subordinate
1 X,α
6 VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
ifandonlyif foranyothersolutionψ ofthe sameequationnotproportionaltoψ
2 1
the following limit property holds:
x ψ (t)2dt
lim 0 | 1 | =0.
x→+∞ x ψ (t)2dt
R0 | 2 |
We will use the following propositionsto find locationof the absolutely continuous
R
spectrum.
Proposition 2.1. [SCS94, Proposition 7] Let Hκ,X,α be the self-adjoint operator
correspondingtothediscretesetX andthesequenceofstrengthsαasinSection2.1.
Assume that for all λ (a,b) there is no subordinate solution for the spectral
⊂
equation τX,αψ =λψ. Then [a,b] σ(Hκ,X,α) and σ(Hκ,X,α) is purely absolutely
⊂
continuous in (a,b).
Proposition 2.2. Let the discrete set X = x : n N on R be ordered as 0<
n +
{ ∈ }
x1 < x2 < .... Assume that it satisfies conditions ∆x∗ := infn∈N xn+1 xn > 0
| − |
and ∆x∗ := supn∈N|xn+1 − xn| < ∞. Let α = {αn}n∈N be a sequence of real
numbers and let λ R. Assume that liminf s √λ > 0. If every solution
n→∞ n
∈
of the equation τ ψ =λψ (see (2.2)) is bounded, then for such λ there exists no
X,α
(cid:12) (cid:0) (cid:1)(cid:12)
subordinate solution. (cid:12) (cid:12)
Proof. Let ψ be an arbitrary solution of τ ψ = λψ. Differentiating (2.4) one
1 X,α
gets in the case s (k)=0
n
6
k (ξ + ξ )
ψ′(x) | | | n| | n+1| for x [x ,x ].
| 1 |≤ s (k) ∈ n n+1
n
| |
There exists N(k) such that infn≥N(k) sn(k) > 0. Since the sequence ξn n∈N
| | { }
is bounded, one has that ψ′(x) is also bounded for x x . Obviously for
1 ≥ N(k)
x x it is bounded too, since it is piecewise continuous with finite jumps at
N(k)
≤
the points of discontinuity. Let ψ be any other solution of τ ψ = λψ, which is
2 X,α
linear independent with ψ . It followsthat there exists a constantC >0 such that
1
(2.7) ψ , ψ , ψ′ , ψ′ C.
k 1k∞ k 2k∞ k 1k∞ k 2k∞ ≤
The Wronskian of the solutions ψ and ψ is independent of x:
1 2
(2.8) W ψ ,ψ :=ψ (x)ψ′(x) ψ′(x)ψ (x), for all x R X.
{ 1 2} 1 2 − 1 2 ∈ +\
This is easy to check: it is constant on every interval (x ,x ) and at the points
n n+1
xn n∈N one has W ψ1,ψ2 (xn+)=W ψ1,ψ2 (xn ) from (2.2). The Wronskian
{ } { } { } −
is non-zero since the solutions are linear independent. From (2.8) one has:
W ψ ,ψ C(ψ (x) + ψ′(x)), for all x R X,
| { 1 2}|≤ | 1 | | 1 | ∈ +\
and therefore there exist constants C and C∗ such that
∗
(2.9) 0<C ψ (x) + ψ′(x) C∗, for all x R X.
∗ ≤| 1 | | 1 |≤ ∈ +\
Now we apply the trick used in the proof of [S92, Lemma 4]. We consider for
an arbitrary n N the interval [x ,x ]. Since the function ψ is continuous on
n n+1 1
∈
[x ,x ], the formula
n n+1
xn+1
(2.10) ψ (x ) ψ (x )= ψ′(t)dt
1 n+1 − 1 n 1
Zxn
holds. Set
∆x
∗
p := .
∗
∆x +2
∗
Next we show that there exists a point x∗ [x ,x ] such that ψ (x∗) p C .
n ∈ n n+1 | 1 n |≥ ∗ ∗
Let us suppose that such a point does not exist, i.e. ψ (x) < p C for all x
1 ∗ ∗
| | ∈
[x ,x ]. We getfrom(2.9)that ψ′(x) >(1 p )C foreveryx [x ,x ]. In
n n+1 | 1 | − ∗ ∗ ∈ n n+1
WIGNER VON-NEUMANN PERTURBATION OF THE KRONIG-PENNEY MODEL 7
particular ψ′ is sign-definite in [x ,x ], so using (2.10) and ∆x ∆x we get
1 n n+1 n ≥ ∗
a contradiction
2p C > ψ (x ) + ψ (x ) ψ (x ) ψ (x ) =
∗ ∗ 1 n+1 1 n 1 n+1 1 n
| | | |≥| − |
xn+1
= ψ′(t)dt>∆x (1 p )C ∆x (1 p )C =2p C .
| 1 | n − ∗ ∗ ≥ ∗ − ∗ ∗ ∗ ∗
Zxn
Thus the point x∗ with required properties exists.
n
Since ψ′(x) C, for every x [x ,x ] such that x x∗ p∗C∗ one has
| 1 | ≤ ∈ n n+1 | − n| ≤ 2C
ψ (x) p∗C∗. Wehaveshownthateveryinterval[x ,x ]containsasubinterval
| 1 |≥ 2 n n+1
of length l :=min ∆x ,p∗C∗ , where ψ (x) p∗C∗. Therefore
∗ 2C | 1 |≥ 2
(cid:0) (cid:1)xn ψ (t)2dt lp2∗C∗2n.
1
| | ≥ 4
Z0
On the other hand,
xn
ψ (t)2dt ∆x∗C2n.
1
| | ≤
Z0
Summing up, for every solution ψ the integral xn ψ(t)2dt has two-sided linear
0 | |
estimate. Thus no subordinate solution exists. (cid:3)
R
2.4. Benzaid-Lutz theorems for discrete linear systems in C2. The results
of [BL87] translate classical theorems due to N. Levinson [L48] and W. Harris
and D. Lutz [HL75] on the asymptotic integration of ordinary differential linear
systemstothecaseofdiscretelinearsystems. Themajoradvanceofthesemethods
that they allow to reduce under certain assumptions the asymptotic integration of
generaldiscrete linear systems to almost trivial asymptotic integrationof diagonal
discrete linear systems. For our applications it is sufficient to formulate Benzaid-
Lutz theorems only for discrete linear systems in C2. The first lemma of this
subsection is a direct consequence of [BL87, Theorem 3.3].
Lemma 2.3. Let µ± C 0 be such that µ+ = µ− and let Vn n∈N
ℓ2(N,C2×2). If the coeffi∈cient\m{a}trix of the discre|te l|in6ear| sys|tem { } ∈
µ 0
−→un+1 = 0+ µ +Vn −→un
−
(cid:20)(cid:18) (cid:19) (cid:21)
is non-degenerate for every n ∈ N, then this system has a basis of solutions −→u±n
with the following asymptotics:
n
1
−→u+n = 0 +o(1) (µ++(Vk)11) as n→∞,
(cid:20)(cid:18) (cid:19) (cid:21)k=1
Y
n
0
−→u−n = 1 +o(1) (µ−+(Vk)22) as n→∞,
(cid:20)(cid:18) (cid:19) (cid:21)k=1
Y
where by (V ) and (V ) we denote the diagonal entries of the matrices V , and
k 11 k 22 k
the factors (µ +(V ) ) and (µ +(V ) ) should be replaced by 1 for those values
+ k 11 − k 22
of the index k for which they vanish.
The following lemma is a simplification of [BL87, Theorem 3.2].
Lemma 2.4. Let tn n∈N ℓ2(N) and Vn n∈N ℓ2(N,C2×2)satisfy thefollowing
{ } ∈ { } ∈
conditions.
(i) Levinson condition: there exists M such that for every n n
− 1 2
≥
n2
1 t <M
n −
| − |
nY=n1
8 VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
and either there exists M such that for every n n
+ 1 2
≥
n2
1+t >M
n +
| |
nY=n1
or
n
1+t as n .
k
| |→∞ →∞
k=1
Y
∞
(ii) The sum V is conditionally convergent and
n
n=1
P
∞
V ℓ2(N,C2×2).
k
( ) ∈
kX=n n∈N
If for every n N
∈
1+t 0
det n +V =0,
0 1 tn n 6
(cid:20)(cid:18) − (cid:19) (cid:21)
then the discrete linear system
1+t 0
−→un+1 = 0 n 1 t +Vn −→un
n
(cid:20)(cid:18) − (cid:19) (cid:21)
has a basis of solutions −→u±n with the following asymptotics:
n
1
−→u+n = 0 +o(1) (1+tk) as n→∞,
(cid:20)(cid:18) (cid:19) (cid:21)k=1
Y
n
0
−→u−n = 1 +o(1) (1−tk) as n→∞,
(cid:20)(cid:18) (cid:19) (cid:21)k=1
Y
where the factor (1 t ) should be replaced by 1 for those values of the index k for
k
−
which it vanishes.
2.5. Characterization of spectra for operators with point interactions via
Weyl function. In this section we view the operator Hκ,X,α as a self-adjoint
extension of a certain symmetric operator and we parametrize this self-adjoint
extension via suitable ordinary boundary triple [Ko75, Br76]. As a benefit of
this parametrization we reduce the spectral analysis of the self-adjoint operator
Hκ,X,α to the spectral analysis of an operator-valued function in the auxiliary
Hilbert space. The reader is referred to [DM95, BGP08] for an extensive dis-
cussion of ordinary boundary triples and their applications. The Weyl function
method was applied to Schr¨odinger operators with δ-interactions by many authors
[Ko89, M95, KM10, GO10, AKM10].
Let X = x : n N be a discrete set satisfying the condition (2.1) and let κ
n
{ ∈ }
be a parameter in the interval [0,π). Define the minimal symmetric operator via
its action and domain
Hκ,Xψ = ψ′′,
−
cos(κ)ψ(0)=sin(κ)ψ′(0)
domHκ,X = ψ: ψ,ψ′′ L2(R+),ψ,ψ′ ACloc(R+ X), ψ(xn−)=ψ(xn+)=0, n∈N, .
(cid:26) ∈ ∈ \ ψ′(xn−)=ψ′(xn+), n∈N. (cid:27)
It can be easily verified that Hκ,X is a closed symmetric densely defined operator
with the adjoint (the maximal operator)
H∗ ψ = ψ′′,
κ,X −
domH∗ = ψ: ψ,ψ′′ L2(R ),ψ,ψ′ AC (R X), cos(κ)ψ(0)=sin(κ)ψ′(0) .
κ,X ∈ + ∈ loc +\ ψ(xn+)=ψ(xn−), n∈N.
n o
WIGNER VON-NEUMANN PERTURBATION OF THE KRONIG-PENNEY MODEL 9
Definition 2.5. For the discrete set X and the parameter κ as above we define
Γ0,κ,X and Γ1,κ,X as
Γ0,κ,X: domHκ∗,X →ℓ2(N), Γ0,κ,Xψ := −ψ(xn) n∈N,
Γ1,κ,X: domHκ∗,X →ℓ2(N), Γ1,κ,Xψ :=(cid:8)ψ′(xn+)(cid:9)−ψ′(xn−) n∈N.
By [BGP08, Proposition 3.17] the triple Πκ,X(cid:8):= ℓ2(N),Γ0,κ,X,Γ(cid:9)1,κ,X is an
ordinary boundary triple for H∗ in the following sense:
κ,X (cid:8) (cid:9)
(i) ran Γ0 =ℓ2(N) ℓ2(N);
Γ1 ×
(ii) for a(cid:16)ll f(cid:17),g domH∗
∈ κ,X
Hκ∗,Xf,g L2 − f,Hκ∗,Xg L2 = Γ1,κ,Xf,Γ0,κ,Xg ℓ2 − Γ0,κ,Xf,Γ1,κ,Xg ℓ2.
Rem(cid:0)ark 2.6. (cid:1)Theas(cid:0)sumption(cid:1)inf x(cid:0) x >0isess(cid:1)ential(cid:0). Inarecentwork(cid:1)[KM10]
n+1 n
n∈N| − |
anordinaryboundarytriplewasconstructedforaclassofsetsX notsatisfyingthis
condition.
For further purposes we define the set σκ,X := σ(Hκ∗,X ↾ kerΓ0,κ,X). We asso-
ciate the Weyl function with the triple Πκ,X
−1
(2.11) Mκ,X(λ):=Γ1,κ,X Γ0,κ,X ↾ker(Hκ∗,X −λI)
which is defined for all λ C σκ,X(cid:16). (cid:17)
∈ \
Recall that s (k)=sin(k(x x )) and c (k)=cos(k(x x )). Given a
n n+1 n n n+1 n
vectorξ ℓ2(N)wecanfindasolut−ionψ ker(H∗ λI)suchtha−tψ(x )= ξ .
∈ ∈ κ,X− n − n
The operatorMκ,X(λ) maps ξ to the vector ψ′(xn+0) ψ′(xn 0) n∈N ℓ2(N).
{ − − } ∈
A straightforward calculation based on the formula (2.4) gives us the values of all
entries of the matrix Mκ,X(λ) except the first row. To find the values of the first
row entries one needs to know ψ(x) on the interval [0,x ], i.e., to solve the non-
1
homogenious Sturm-Liouville problem ψ′′(x)=λψ(x), ψ(0)cosκ ψ′(0)sinκ =
− −
0,ψ(x )= ξ . The result of calculations is the following:
1 1
−
b (k) a (k) 0 ...
1 1
a (k) b (k) a (k) ...
1 2 2
Mκ,X(k2)= 0 a2(k) b3(k) ...,
 ... ... ... ...
 
 
with
c (k)cosκ ks (k)sinκ c (k)
0 0 1
b (k):=k − +k ,
1 s (k)cosκ+kc (k)sinκ s (k)
0 0 1
c (k) c (k)
n n−1
b (k):=k + , n 2,
n
sn(k) sn−1(k)! ≥
k
a (k):= .
n
−s (k)
n
Remark 2.7. Notethatthesetσκ,X consistsofsquaresofallzeroesofdenominators
in the formulas above:
(2.12)
σκ,X = k2 R:sn(k)=0 for some n N or s0(k)cosκ+kc0(k)sinκ =0 .
{ ∈ ∈ }
Letα= αn n∈N be a boundedsequence ofrealnumbers. Define anoperatorin
ℓ2(N) { }
(2.13) Bαξ = αnξn n∈N, ξ ℓ2(N),
{− } ∈
10 VLADIMIRLOTOREICHIKANDSERGEYSIMONOV
which is bounded and self-adjoint. The operator Hκ,X,α is a restriction of the
operator H∗ corresponding to B in the following sense
κ,X α
Hκ,X,α =Hκ∗,X ↾ker(Γ1,κ,X −BαΓ0,κ,X).
Proposition 2.8. [BGP08, Theorem 3.3] Let the discrete set X, the sequence of
strengths α and the self-adjoint operator Hκ,X,α be defined as in Section 2.1. Let
σκ,X be the set defined in (2.12) and Bα be given by (2.13). Let Mκ,X be the Weyl
function (2.11)associatedwiththeordinary boundarytripleΠκ,X. Forλ R σκ,X
∈ \
the following holds:
λ σess(Hκ,X,α) if and only if 0 σess Bα Mκ,X(λ) .
∈ ∈ −
3. Spectral and asymptotic a(cid:0)nalysis (cid:1)
3.1. Asymptotic analysis of a special class of discrete linear systems. In
this section we study a special class of discrete linear systems that encapsulates
system (2.5) corresponding to X = xn: n N and α = αn n∈N as in Models I
{ ∈ } { }
and II described in the introduction.
Let the parameters l, a, b, γ and ω satisfy conditions
1 π
(3.1) l R , a,b C, γ ,1 and ω 0, .
+
∈ ∈ ∈ 2 ∈ 2
(cid:18) (cid:21) (cid:16) (cid:17)
For further purposes we define
(3.2) µ :=l l2 1,
±
± −
ae−iω+be−2iω p
(3.3) z := , β := z , ϕ:=argz,
2isinω | |
and
exp βn1−γ , if γ <1,
(3.4) f±(β):= ± 1−γ
n ( n±β(cid:16), (cid:17) if γ =1.
Thefollowinghastechnicalnatureandhelpstosimplifytheanalysisofcases(Model
I and Model II).
Lemma 3.1. Let the parameters l, a,b, γ and ω be as in (3.1). Let µ , β,ϕ and
±
f±() be as in (3.2), (3.3) and (3.4), respectively. Let the sequence of matrices
nRn· n∈N ℓ1(N,C2×2 . If the coefficient matrix of the discrete linear system
{ } ∈ }
0 1 0 0 e2iωn 0 0 e−2iωn
(3.5) −→un+1 = 1 2l + a b nγ + a b nγ +Rn −→un
(cid:20)(cid:18)− (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
is non-degenerate for every n ∈ N, then this system has a basis of solutions −→u±n
with the following asymptotics.
(i) If l R cosω,1 , then
+
∈ \{ }
1
−→u±n = µ± +o(1) µn± as n→∞.
(cid:20)(cid:18) (cid:19) (cid:21)
(ii) If l=cosω, then
cos(ωn+ϕ/2)
−→u+n = cos(ω(n+1)+ϕ/2) +o(1) fn+(β) as n→∞,
(cid:20)(cid:18) (cid:19) (cid:21)
and
sin(ωn+ϕ/2)
−→u−n = sin(ω(n+1)+ϕ/2) +o(1) fn−(β) as n→∞.
(cid:20)(cid:18) (cid:19) (cid:21)
Before providing the proof we make several remarks.