zeros of spectral density of von neumann schrodinger

8/11/2019 Zeros of Spectral Density of Von Neumann Schrodinger

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ZEROES OF THE SPECTRAL DENSITY OF THE

PERIODIC SCHRODINGER OPERATOR WITH

WIGNER-VON NEUMANN POTENTIAL

SERGEY NABOKO AND SERGEY SIMONOV

Abstract. We consider the Schrodinger operatorL on the half-line with a periodicbackground potential and the Wigner-von Neumann potential of Coulomb type:c sin(2x+)

x+1 . It is known that the continuous spectrum of the operatorL has the sameband-gap structure as the free periodic operator, whereas in each band of the absolutelycontinuous spectrum there exist two points (so-called critical or resonance) where the op-eratorL has a subordinate solution, which can be either an eigenvalue or a half-boundstate. The phenomenon of an embedded eigenvalue is unstable under the change of theboundary condition as well as under the local change of the potential, in other words,it is not generic. We prove that in the general case the spectral density of the operatorL has power-like zeroes at critical points (i.e., the absolutely continuous spectrum haspseudogaps). This phenomenon is stable in the above-mentioned sense.

1. Introduction

It is well-known that the Schrodinger operator on the half-line with a summable potentialhas purely absolutely continuous spectrum on R+ [28]. It is also known [12] that if thepotential is square summable, then the absolutely continuous spectrum still covers thepositive half-line, see also the work [17]. In this case however there is no guarantee thatthe positive singular spectrum is empty. Moreover, there are explicit examples of potentials[25, 27], which produce dense point spectrum on R+. In a sense, the simplest example ofthe potential producing positive eigenvalues is the so-called Wigner-von Neumann potentialc sin(2x)

x . The self-adjoint operatorL0, given by the differential expression l0 := d2dx2 +

c sin(2x)x

and the boundary condition (0)cos (0) sin = 0, acting in L2(R+) on thedomain

Dom L0, =

L2(R+) H2loc(R+) :l0L2(R+), (0) cos (0) sin = 0

may have an eigenvalue at the point 2 [30], while the rest ofR+ is covered by the purely

absolutely continuous spectrum [2]. Due to the subordinacy theory [13], this can be seenfrom the behavior of generalized eigenvectors (i.e., the solutions of the spectral equation

l0 = ): for R+\{2} there is a base of them with asymptotics eix +o(1) as

x+. For = 2 the base is different: x c4 (sin(x) + o(1)) and x c4 (cos(x) + o(1)).This point is called the resonance (or critical) point due to the change of the type ofasymptotics of generalized eigenvectors. One of the solutions (up to a constant) of theequation l0 =

2 is subordinate. If on top of that it belongs to L2(R+) and satisfiesthe boundary condition at the origin, then it represents an eigenfunction ofL0,. It isclear that the effect of appearance of positive eigenvalue is highly unstable. It happenswith probability one in a suitable sense: the eigenvalue disappears if one changes slightly

1991 Mathematics Subject Classification. 47E05,34B20,34L40,34L20,34E10.Key words and phrases. Asymptotics of generalized eigenvectors, Schrodinger operator, Wigner-von

Neumann potential.1


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ZEROES OF THE SPECTRAL DENSITY 2

the boundary condition or adds a summable (or even compactly supported) perturbationto the potential. If the notion of resonances makes sense for the operator (i.e., if thekernel of the resolvent admits the analytic continuation through the continuous spectrum),this eigenvalue may become a resonance under such perturbation. The set of summablefunctions which one could utilize as perturbations of the potential preserving the eigenvalueat the point 2 was studied in [10]. On the other hand, the phenomenon of the change ofthe type of asymptotics of generalized eigenvectors at a point of the absolutely continuousspectrum is stable under summable perturbations of the potential and does not depend onthe boundary condition. It is therefore meaningful to consider objects which are a moreor less stable in this sense: one can study the Weyl-Titchmarsh function m or the spectraldensity (the derivative of the spectral function of the operator), which are related bythe equality

() = 1

Im m(+i0) for a.a. R.

The behavior of the Weyl function near resonance points has been studied by Hinton-Klaus-

Shaw [15, 18] and Behncke [2, 3, 4] for Schrodinger and Dirac operators with potentials of,in particular, Wigner-von Neumann type.In the present paper we consider the differential Schrodinger operatorL with the po-

tential that is the sum of the following three parts: a periodic backgroundq, a potentialof Wigner-von Neumann type c sin(2x+)

x+1 and a summable partq1. The operator is defined

by the differential expression

(1) l:= d2

dx2+q(x) +

c sin(2x+)

x+ 1 +q1(x)

and the boundary condition (0) cos (0)sin = 0, [0; ). It acts in L2(R+) ona suitable domain. We assume that the function qhas the period aand is summable over

this period: q L1(0; a). The parameters c and are real constants. The operatorL isthen self-adjoint in L2(R+).

The geometry of the spectrum in the case of periodic background differs from the caseq(x)0. It is well-known (see, e.g., [22]) that the absolutely continuous spectrum of theoperator given by the expression l on the whole real line coincides with the spectrum ofthe corresponding periodic operator on the whole real line,

(2) Lper = d2

dx2+q(x),

i.e., has a band-gap structure:

(Lper) =:j=0

([2j; 2j] [2j+1; 2j+1]),

where 0 < 0 1 < 1 2 < 2 3 < 3 4 < ... It is clear that the absolutelycontinuous spectrum ofL coincides set-wise with (Lper), although (Lper) has multi-plicity two, whereas (L) is simple; note thatL is considered on the half-line. In everyband there exist two resonance points j,+ and j,. The type of asymptotics of gener-alized eigenvectors at these points is different from that in other points of the absolutelycontinuous spectrum. Subordinate solution appears and thus each of the resonance pointscan be an eigenvalue of the operatorL as long as this solution satisfies the boundarycondition and belongs toL2(R+). The exact locations of the pointsj, are determined by

the quantization conditions [22]k(j,+) =

j+ 1

a

, k(j,) =

j+

a

, j0,


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where k() is the quasi-momentum of the periodic operatorLper and{} is the standardfractional part function. The above-mentioned condition / Z

2a guarantees that resonance

points firstly do not coincide with the band boundaries and secondly that they do not glueup together.

The main goal of the present paper is to analyze the interplay between the periodicstructure and the singular Wigner-von Neumann perturbation. The main result is The-orem 2 of Section 5, which essentially tells us that the spectral density of the operatorLhas power-like zeroes at each of the resonance points. Let +(x, ) and (x, ) be theBloch solutions of the periodic equation(x) +q(x)(x) =(x). Let (x, ) be thesolution of the Cauchy problem

(3) l=, (0) = sin , (0) = cos .

Denote by W{+(), ()} = W{+(, ), (, )} the Wronskian of two Bloch solu-tions.

Theorem. Letq1L1(R+), / Z

2a , and() be the spectral density of the operatorL.Let the indexj 0. If is such that the solution(x, j,+) of (3) is not a subordinateone, then there exist two non-zero limits

limj,+0

()

| j,+|2|c|

a|W{+(j,+),(j,+)}|

a

02+(t,j,+)e

2itdt

.

Analogously, if is such that(x, j,) is not subordinate, then there exist two non-zerolimits

limj,0

()

| j,|2|c|

a|W{+(j,+

),(j,+

)}|

a

0

2(t,j,)e2itdt

.

Note that the condition of non-subordinacy of is indeed satisfied in the generic case.Consider one of the critical points cr. There is a unique [0; ) for which issubordinate, denote it by cr. Denote also :=

|c|a|W{+(cr),(cr)}|

a0

2(t, cr)e2itdt

where the choice of sign coincides with the one in cr = j,. It follows from [22] andthe analysis below that cr(x, cr) =x

(c++(x, cr) +c(x, cr) +o(1)) asx+where +, are the Bloch solutions of the periodic equation(x) + q(x)(x) =(x)and c+, c are some constants. Therefore cr is an eigenvalue ofLcr iff > 12 . At thesame time, for > 1

2 and

=cr the order of the zero of

() at the point cr is greater

than one. This fact is in exact correspondence with the result of Aronszajn-Donoghue [1],by which if the point is an eigenvalue of the operatorLcr , then the integral

R

d()()2 is

finite for every =cr.It should be mentioned, that this result in the particular case of zero background pe-

riodic potential q follows from the work of Hinton-Klaus-Shaw [15]. They considered thedifferential Schrodinger operator on the half-line with the potential, which is an infinitesum of terms of Wigner-von Neumann type plus a rapidly decreasing term analogous to ourq1 but decaying faster. They studied the behavior of the Weyl-Titchmarsh function nearthe critical points in the case = cr and 0 was a by-product of their analysis. Nevertheless, the mostessential part of the problem in their work is similar to the one in our case. However, ourapproach is different from that of papers [15, 18], see the detailed discussion in Section 6.


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Our goal was to elaborate a more general approach (reduction to a model problem), whichwould enable us to apply it without any significant changes to different operators.

Zeroes of the spectral density of the Schrodinger operator divide the absolutely continu-ous spectrum of the operator into independent parts. This phenomenon is called a pseudo-gap and has a clear physical meaning. Eigenvalues embedded into the continuous spectrumhave been observed in experiment [9]. Operators with Wigner-von Neumann potentials at-tracted attention of many other authors, e.g., [7, 8, 24, 15, 18, 4, 3, 2, 21, 20, 22, 26].

The paper is organized as follows. In Section 2, we state the Weyl-Titchmarsh typeformula for the operatorL which was proved in [23] and serves as our main tool in thework with the spectral density. In Section 3, we write the spectral equation for the operatorL in an equivalent form of a discrete linear system and restate the Weyl-Titchmarsh typeformula in terms of the asymptotic behavior of solutions of that system. In Section 4,we study this discrete system (which can be regarded as a model problem). This can beviewed as an independent task. In Section 5, we use the results of the preceding analysisto find the asymptotics of the spectral density of the operatorL near the critical points.In Section 6, we make final comments characterizing our method in comparison to the oneutilized in the work [15] and give a discussion of a few examples of its applicability.

2. Preliminaries

Spectral properties of second-order differential Schrodinger operators are related to theasymptotic behavior of their generalized eigenvectors [13]. One of illustrations of thisrelation is the Weyl-Titchmarsh type formula for the spectral density in terms of asymptoticcoefficients of the solution to the spectral equation that satisfies the boundary condition.The spectral equationl = can be considered as a perturbation of the periodic equation

(x) +q(x)(x) =(x).Let us denote by :={j, j , j 0} the generalized boundary of the spectrum of

the periodic operatorLper (it can be not exactly the boundary of the set (Lper), becausethe endpoints of the neighboring bands can coincide). We will use the following result(Weyl-Titchmarsh type formula).

Proposition 1 ([23]). Let / Z2a

and q1 L1(R+). Then for every fixed (Lper)\( {j,+, j,, j 0}) there exists a non-zero constant A() dependingon such that

(4) (x, ) =A()(x, ) +A()+(x, ) +o(1) asx+,

(x, ) =A()(x, ) +A()+(x, ) +o(1) asx+,and the following equality holds:

(5) () = 1

2|W{+(), ()}| |A()|2 ,

where() is the spectral density of the operatorL.This result is a generalization of the classical Weyl-Titchmarsh (or Kodaira) formula

[29, 19]. A variant of this formula for the Schrodinger operator with Wigner-von Neumann

potential without the periodic background (q(x)0) follows from the results of [24, 6]. Inthe case of discrete Schrodinger operator with Wigner-von Neumann potential an analogousformula is also known, see [11, 16].


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3. Discretization

In this section we pass over from the spectral equationl = to a specially constructeddiscrete linear system essentially equivalent to l = and rewrite the Weyl-Titchmarshtype formula expressing the spectral density in terms of solutions of this system.

Step 1. Variation of parameters and discretization: perturbation of the mon-

odromy matrix. First we perform a transformation of the spectral equation to the differ-ential system in order to get rid of the periodic background potential (in fact, this simplyis a variation of parameters). Consider (Lper)\and equation l =. Define newvector-valued function (x) by the following equality:

(x)(x)

=

(x, ) +(x, )(x, )

+(x, )

(x).

By a straightforward substitution one has:

(6) (x) =L(x, )(x),

where

L(x, ) :=

c sin(2x+)x+1

+q1(x)

W{+(), ()}+(x, )(x, ) 2+(x, )

2(x, ) +(x, )(x, )

.

Denote by (x0, x , ) the fundamental matrix for the system (6) (i.e., the matrix solution of(6) satisfying (x0, x0, )I). Denote also byMn() the monodromy matrix correspond-ing to the shift by the period a of the background potential, Mn() := (a(n 1),an,).Instead of the solution (x) of the differential system (6) consider the following sequence{wn}n=1 of vectors from C2:

wn:=(a(n 1)),which obviously solves the discrete linear system wn+1 = Mn()wn. In what follows, wedeal with this system transforming it to a simpler form. Using the standard perturbationmethods (see for example [22]) we can determine the matrix Mn() up to a summable (inn) term.

We have passed from the continuous variable x to the discrete variable n. Later, inSection 4 we will return to continuous variables considering a new one, y. There is nostrict necessity in such a trick. One could think that the discretization determined by theperiod a of the background potential qhelps to get rid of this potential. However, this isnot completely true: this role is taken by the above variation of parameters transformation.One could do simpler: the coefficient matrix of the differential system (6) is the sum oftwo terms. The first term has a factor 1

x+1

multiplied by an infinite sum of the exponentialterms coming from the Fourier decomposition of the periodic parts of Bloch solutions.Each of these terms can resonate (become constant in x) for certain values of. Thishappens exactly at the resonance pointsj,. The second term is a summable matrix-valuedfunction. Non-resonating exponential terms can be eliminated using the uniform Harris-Lutz transformation in a fashion of [23]. This approach would lead to a simple differentialmodel system equivalent to the equation l = . However, we use the approach of thediscretization, which brings some extra technical difficulties and complicates the notation.The reason is that we obtain as a result a discrete model system, which can serve widerneeds. In particular, it is possible to consider the discrete Schrodinger operator withthe discrete Wigner-von Neumann potential which has two critical points on the interval

[2; 2] (covered by the absolutely coninuous spectrum of this operator) and to prove thatits spectral density has zeroes of the power type at the critical points. See Discussion forthe exact formulation of this result. We plan to prove it in a forthcoming paper.


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Let us introduce the following notation for uniformly summable sequences. LetRn() be2 2 matrices for nNand S (whereS is an arbitrary set). We write{Rn()}n=1l1(S), if there exists a sequence of positive numbers{rn}n=1l1 such that for everySand n N,Rn()< rn. We start with a simple asymptotic formula for the monodromymatrix:

Lemma 1. For every(Lper)\,

Mn() =I+ 1

an

ana(n1)

c sin(2t+)

W{+(), ()}

+(t, )(t, ) 2+(t, )

2(t, ) +(t, )(t, )

dt+R(1)n (),

where{R(1)n ()}n=1l1(K) for every compact setK(Lper)\.Note that it follows that for every n N the matrix R(1)n () depends continuously on

(Lper)\.Proof. We give only a sketch of the proof, because it is rather standard. Fundamentalmatrix for (6) satisfies the integral equation

(7) (x0, x , ) =I+

xx0

L(t, )(x0, t , )dt.

To studyMn() putx0 = a(n1) and consider Volterra integral operatorVn() in Banachspace C([0; a], M2,2) (of 2 2 matrix functions, with any matrix norm) defined by the rule

Vn() :u(x) x0

L(a(n 1) +t, )u(t)dt.

Using Neumann series and direct estimate of the norm we can show the existence of (IVn)1 and the following estimate for its norm:

(8) (I Vn())1 exp an

a(n1)L(t, ) dt

.

Returning to (7) we can write the matrix equality

(a(n 1), a(n 1) +x, ) = ((I Vn())1I)(x) ==I+ (Vn()I)(x) + (V2n()(I Vn())1I)(x),

and so putting x= a we have:

Mn() =I+ ana(n1)

L(t, )dt+ (V2n()(I Vn())1I)(a) =:R

(2)n ()

.

The remainder R(2)n () is continuous in (Lper)\ for every n and satisfies the uniform

estimate

R(2)n () =O

ana(n1)

L(t, ) dt2

as n .

Fix the compact set K(Lper)\. One can always choose Bloch solutions so that theirWronskian is analytic functions and does not vanish on (Lper). From the properties ofBloch solutions it follows that there exists c1(K) such that for every

Kand x, 1W{+(), ()} +(x, )(x, ) 2+(x, )2(x, ) +(x, )(x, ) c1(K).


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Step 3. Elimination of non-resonant terms by Harris-Lutz method. For every

fixed (Lper)\the entries of the matrix

(d)n ()

(ad)n ()

(ad)n () (d)n ()

are linear combina-

tions of four exponential terms: e2ian,e2ian,e2i(k()+a)n and e2i(k()a)n. The first two

of them do not depend on and since / Z2a they do oscillate. The third and fourth termsdo depend on. They oscillate, ifk() + a /Z andk()a /Z, respectively, other-wise they are constant (we call resonance this matching of the quasi-momentum k() andthe frequency ). Oscillating exponential terms can be dropped with the help of Harris-Lutz transformation and do not affect the asymptotical type of solutions of the systemwn+1 = Mn()wn. Therefore (see the rigorous proof of this fact below) if is not one ofthe resonance pointsj,, which are defined by conditions

k(j,+) :=j+ 1

a

, k(j,) :=

j+

a

, j0,

then every solution of the system wn+1 =Mn()wn has a limit as n . Note that dueto the condition /

Z

2a , for every j0 resonance points j,+=j, do not coincide withthe endpoints of the j-th spectral band.Pick a spectral band and one of the two critical points in this band, e.g., j,+ for some

index j. To simplify the notation, we writecr := j,+. Consider an open neighborhoodUcr of the point cr such that it contains neither the critical point j, nor the endpointsof the band j and j. In what follows we assume that cr and Ucr are fixed and we dropthe indices j, + in most cases. To remove the oscillating terms from the system we needthe following elementary lemma belonging to the domain of the mathematical folklore.

Lemma 2. For every real /2Z andn N,

m=n

eim

m 1n sin 2 .Proof.(ei 1)

m=n

eim

m

=

m=n

ei(m+1)

1

m 1

m+ 1

e

in

n

m=n

1

m 1

m+ 1

+

1

n=

2

n.

Since|ei 1|= 2sin

2, this argument completes the proof.

By a Harris-Lutz transformation uniform in Ucr it is possible to eliminate non-res-onating exponential terms and to stabilize coefficients at resonating terms (i.e., make themindependent of). Doing this we can provide for the uniform summability of the remainder.We formulate this argument in the following lemma.

Lemma 3. There exists a sequence of matrices{Q(1)n ()}n=1 defined in Ucr such thatQ

(1)n () =O

1n

asn uniformly inUcr and

(15) expQ(1)n+1()

Mn()exp

Q(1)n ()

=I+

1

n 0 +(cr)e

2i(k()+a)n

+(cr)e2i(k()+

a)n

0 +R(3)n ()with some{R(3)n ()}n=1l1(Ucr) such thatR(3)n () is continuous inUcr for everyn.


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Proof. We follow the scheme of [5] and need to take care of the uniformity only. The

explicit formula for Q(1)n is

(16) Q(1)n () :=

m=n1

m

(d)m ()

(ad)m () +(cr)e2i(k()+a)m

(ad)

m () +(cr)e2i(k()+a)m

(d)

m () .

Lemma 2 yields immediately the estimates

m=n

(d)m ()

m

2|0()|n| sin a|and

m=n

(ad)m () +(cr)e2i(k()+a)

m

|+() +(cr)|

n| sin(k() +a)| + |()|

n| sin(k() a)| .

Functions 0 and are continuous and the denominators sin a and sin(k()

a) are

separated from zero in Ucr. On the other hand, the denominator sin(k() +a) has theonly zero at the point cr, which is simple and thus is compensated by the zero of the

numerator. Therefore the estimate Q(1)n () =O1n

holds and is uniform in Ucr. Using the

obvious property

Q(1)n+1() Q(1)n () = 1

n

(d)n ()

(ad)n () +(cr)e2i(k()+a)n

(ad)n () +(cr)e2i(k()+a)n (d)n ()

,

one obtains:

Mn() =I+1

n 0 +(cr)e

2i(k()+a)n

+

(cr)e2i(k()+a)n 0

+Q(1)n+1() Q(1)n () +R(1)n ().

Multiplying by exp(Q(1)n ()) from the right; by exp(Q(1)n+1()) form the left, expanding

exponents and absorbing the terms of the order 1/n2 in the remainder, we have:

expQ(1)n+1()

Mn()exp

Q(1)n ()

=I+

1

n

0 +(cr)e

2i(k()+a)n

+(cr)e2i(k()+a)n 0

+R(1)n () +O

1

n2

=R(3)n (),

where the estimate O 1n2

is uniform in Ucr. It is clear thatR

(3)n () is continuous in Ucr

for every n. This completes the proof.

Step 4. Reduction to the model problem. After the Harris-Lutz transformation{wn}n=1 {wn}n=1 withwn:= exp(Q(1)n ())wn the system wn+1 = Mn()wn is reduced to the system

wn+1=

I+

1

n

0 +(cr)e

2i(k()+a)n

+(cr)e2i(k()+a)n 0

+R(3)n ()

wn.

At the critical point it takes the following form:

wn+1= I+ 1n

0 +(cr)+(cr) 0

+R(3)n (cr)

wn.


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It is convenient to diagonalize the constant matrix in the second term. One has: e

i2 arg +(cr) ie

i2 arg +(cr)

ei2arg +(cr) ie i2 arg +(cr)

1 0 +(cr)

+(cr) 0

e i2 arg +(cr) ie i2 arg +(cr)

ei2 arg +(cr) iei2 arg +(cr)

=|+(cr)| 1 00 1 .Consider the following sequence{vn}n=1 instead of{wn}n=1:

vn:=

e

i2arg +(cr) ie

i2arg +(cr)

ei2arg +(cr) ie i2arg +(cr)

1 wn.Due to (15) the system wn+1 = Mn()wn is equivalent to:

(17) vn+1=

I+

n

cos(2(k() +a)n) sin(2(k() +a)n)sin(2(k() +a)n) cos(2(k() +a)n)

+R(4)n ()

vn,

where the sequence{R(4)n ()}n=1l1(Ucr), the functionsR(4)n () are continuous in Ucr and

:=|+(cr)|= |c|2a|W{+(cr), (cr)}|

a0

2+(t, cr)e2itdt

.Replace the parameter on the set Ucr by the new small parameter

(18) := 2(k() k(cr)) = 2(k() +a) 2j+ 1 +

a

,

where is the standard floor function. The set of values taken by is U :={2(k() k(cr)), Ucr}. By the property of the quasi-momentum (thatk() [j ; (j+ 1)] inj-th spectral band) and the condition /

Z

2a

, which guarantees that the critical point is

in the interior of the spectral band, we have U (2; 2). DenoteRn() =R(4)n () forcorresponding to according to (18). System (17) then reads:

(19) vn+1=

I+

n

cos(n) sin(n)sin(n) cos(n)

+Rn()

vn,

The aim of this section is to rewrite the Weyl-Titchmarsh type formula in terms of thesolutions of the system (19). Proposition 1 deals with the solution of the spectralequation for the operatorL. Combining all the transformations described above anddenoting the result by : (x) {vn}n=1, one ends up with the following model imageof the initial solution :

(20) v,n() =v,n(2(k() k(cr))) :=

ei2arg +(cr) ie

i2 arg +(cr)

ei2arg +(cr) iei2arg +(cr)

1 expQ(1)n () (a(n 1), ) +(a(n 1), )(a(n 1), ) +(a(n 1), )

1 (a(n 1), )(a(n 1), )

.

This is obviously a solution of the system (19) and it is continuous in U for every n.

Lemma 4. For everyU\{0} there exists a limit limn

v,n()= 0, which is continuousinU\{0} as a vector-valued function of. The spectral density ofL equals

(21) () = 2

|W{+(), ()}|limn

v,n(2(k() k(cr)))2 a.e. inUcr.


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Proof. A straightforward substitution of the transformation yields this result. Indeed,asymptotics of the solution and of its derivative (4) together with the boundedness of

(x, ) +(x, )(x, )

+(x, )

1imply that there exists the limit

limx+

(x, ) +(x, )(x, )

+(x, )

1 (x, )(x, )

= A()A()

.Furthermore, since exp

Q(1)n ()I as n , there exists the limitlimn

v,n() =1

2

1 1i i

e

i2arg +(cr)A()

ei2arg +(cr)A()

.

It follows that |A()|2 =limn

v,n()2 /4, and substitution of this to the Weyl-

Titchmarsh type formula (5) completes the proof.

The summary of this section is that the study of the spectral density ofLis reduceable tothe study of the system (19) and, more precisely, to the study of the behavior of limn

v,n()

for small.In the general case (cr = j,) denote the coefficients , which may be different at

different resonance points, asj,. Explicit calculations give: j,= c a0 2(t,j,)e2itdt2aW{+(j,),(j,)}.

Remark 1. Coefficientsj, are not necessarily non-zero, because they are proportionalto the Fourier coefficients of p+(, j,), which might be zero. E.g., consider the case ofzero periodic potential, q(x)0. In this case one can choose the perioda arbitrarily andthe result is independent of the choice (except for the case Z

a ). For any fixeda, the

half-line is divided into spectral bands with coinciding endpoints, ja 2; (j+1)a 2, j0.The quasi-momentum is k() = a

, the critical points are j,+ =

a

j+ 1 a

2andj, =

a

j+

a

2, j 0. Bloch solutions are(x, ) = eik()xa = e

x and

their periodic parts p+(x, ) p(x, ) 1 have only one non-zero Fourier coefficient.Explicit calculation shows thatj,+ 0 andj, = 0 for every j=a. For the singleexisting resonance point one has: a

, = 2 and a

, =

|c|4

. This coincides with theclassical results on the Wigner-von Neumann potential [30]. Our result concerning zeroesof the spectral density in this case is in accordance with the result of Hinton-Klaus-Shaw[15].

4. Model problemIn this section our aim is to study the dependence on , which can be arbitrarily small,

of the limits of solutions to the system (19). As we have shown in the previous section,this is equivalent to the study of the behavior of the spectral density ofL near criticalpoints.

Let us make few comments on the structure of the coefficient matrix of the model system.

One can write it as follows: I+ n

Dn3+Rn(), where D =

cos sin sin cos

is the

matrix of rotation by the angle and 3 =

1 00 1

(notation for the Pauli matrix)

is the reflection matrix. The presence of the latter plays a very important role. With3,the system is elliptic for U\{0} and hyperbolic for = 0 (we call the system elliptic,if it has a base of solutions of the same order of magnitude and hyperbolic in the opposite


12/24


case). Without3, there is no change of type of the system at the point = 0, the systemis always elliptic. If 3 is absent, one can factor out the diagonal term of the first two

summands 1 +cos(n)n

and the coefficient matrix reduces to I+ sin(n)n

0 11 0

+Rn()

with some other uniformly summable sequence{

Rn()}n=1

. The matrix in the second termhere is constant and diagonalizable. Its spectrum is purely imaginary, and the Levinsontheorem immediately gives the answer (=asymptotics of solutions), which is uniform in U including the point = 0. We could say that the problem is scalarized in thiscase. The situation is different in our case, since the difference between the two eigenvaluesis not pure imaginary:

I+

nDn3

=

1 n

. This leads to serious troubles in the

analysis and exhibits a new phenomenon.One may expect that for sufficiently small values of the magnitude of solutions is

determined mostly by diagonal elements 1 cos(n)n

. We want to transform the system ina way such that for every U (including = 0) the limit of every solution as n exists (for the system (19) this is not true: if = 0, then one of the solutions grows asn).

To this end, we make the following substitution: vn = expn1 cos(r)r drun. This leadsto the system un+1 = Bn()un with

(22) Bn() := exp

n+1n

cos(r)

r dr

I+

n


+Rn()

.

The existence of the limit of any solution of the system un+1=Bn()unis equivalent to theconvergence of the infinite product

n=1Bn() (in fact for = 0 this follows from Lemma

4). Moreover, most of the statements that we make about the asymptotic behavior ofsolutions of the systemun+1=Bn()un can be formulated in terms of products of matricesBn(). In what follows we will choose the way of formulation depending on the convenience

of its use. We are going to work with particular solutions determined by fixing their initialvalues, this is a discrete analogue of the Cauchy problem. To this end, we introduce thefollowing notation: for given Uand the vector of initial data f C2 define the vectorsequence{un(, f)}n=1 by the recurrence relation

(23) u1(, f) :=f,

un+1(, f) :=Bn()un(, f), n1.Note that due to the decomposition of the exponential term the matrix Bn() can be

written as

Bn() =I+

n cos(n)

n+1

n cos(r)dr sin(n)

sin(n)

cos(n)

n+1

n cos(r)dr

+

Rn()

with a sequence{ Rn()}n=1l1(U). One can rewrite the system un+1 = Bn()un as(24) un+1 un=

n

cos(n) n+1

n cos(r)dr sin(n)

sin(n) cos(n) n+1n

cos(r)dr

+Rn() un.

The behavior of the solutions can be observed in a scale of the variable y = n|| (slowvariable). If one puts z(y,,f) := u y||(, f) and divides by||, then (24) becomesapproximately

(25) z(y+ ||) z(y)

||

sign

y

0 sin ysin y 2cos y

+

1

||R y||()

z(y)

The remainder 1|| R y||() is a step-wise constant matrix-valued function which is com-pressed in 1|| times in the horizontal scale and stretched in

1|| times in the vertical scale,


13/24


therefore it concentrates near the origin of the variable y and its L1 norm is preserved. Sowe may expect that the remainder term will be absorbed into a new boundary conditionof the limit problem. Expression on the left hand side of (25) becomes the derivative as0, and one obtains the following equation for the limits h(y, f) := lim

0z(y,,f):

(26) h(y) =y 0 sin y

sin y 2cos y

h(y),

cf. (54). The remainder 1||R y||() plays a role only in determining the initial values

h(0) for the solutions of (26). It even suffices to know onlyRn(0) to determine this initialvalue, under the condition of continuity ofRn() for everyn. Thus the picture of the wholephenomenon can be described as follows: there exist two scales: fast, discrete,n N,and slow, continuous, y R+. The system first moves along the first (fast) scalewith = 0. The limit of the solution as n for = 0 serves as initial value for thedifferential equation in the second (slow) scale. Our aim in this section is to prove the

following result which gives the exact formulation of the above considerations. We preferto write an integral equation in the slow variable instead of the differential one, because thefirst has a unique solution while with the second one can have troubles due to the differentbehavior of the solutions near the origin in different cases depending on the value of.

Theorem 1. Assume that functions Rn() are continuous in U for every n N, thematricesBn() are invertible for everyn N, Uand the sequence{Rn()}n=1l1(U).Then for everyy >0 andf C2 there exist two limits

h(y, f) := lim0

u y||(, f),which satisfy the following integral equations:

(27) h(y, f) = limn

un(0, f) y0

0 1exp

2yt

cos ss

ds

0

sin tt

h(t, f)dt.

Moreover, the following four limits exist and are equal:

(28) lim0

limn

un(, f) = limy+

h(y, f).

Additionally, the linear map :f limn

un(0, f) has rank one.

Remark 2. 1. It follows that the linear map f lim0

limn

un(, f) also has rank one.

2. Note that limn

un(0, f) = limn

lim

0un(, f) and lim

0limn

un(, f) are the limits of the

same expression taken in the different order and that they do not coincide. Our aim is toprove that the second limit of these two exists. It will fol low then that the spectral densityofL can have zeros at critical points. In fact we can rewrite the second expression aslim0

limy+

u y||(, f). Here the convergence of the first limit is uniform in (unlike theconvergence of lim

nun(, f) ), and this makes it possible to change the order of limits as

in (28).

Proof. We divide the proof of Theorem 1 into four steps.

Step I. A priori estimate and uniform convergence of the tail for the matrix

product. We start with few technical results concerning the system un+1 = Bn()un.These include uniform boundedness of its solutions (in both variables, n and ) and theuniform with respect to convergence in the slow variable y.


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Lemma 5. LetBn() be given by (22) and{Rn()}n=1l1(U). Then for everyU theproduct

n=1Bn()

converges. If matrices Bn() are invertible for every n N and U, then for everynon-zero the product is invertible, while for zero it is of rank one.

Proof. This follows from the discrete Levinson theorem [5]. Using the expansion

(29) exp

n+1n

cos r

r dr

= 1

n

n+1n

cos(r)dr+O

1

n2

as n ,

we rewrite each matrix of the sequence Bn() in the following form (at places, we dropthe argument in order to simplify the notation, hoping that this will not lead to anyconfusion):

(30) Bn=I+V(1)n +R

(5)n ,

where{R(5)n ()}n=1l1(U) and

(31) V(1)n () :=

n

cos(n) n+1

n cos(r)dr sin(n)

sin(n) cos(n) n+1n

cos(r)dr

.

One has:

(32)

n+1n

cos(r)dr=cos 1

sin(n) +

sin

cos(n).

For = 0 the conditions of Theorem 3.1 from [5] are satisfied. The named theoremyields the existence of a base of solutions of the system un+1 = Bn()un, which have the

asymptotics 10

+o(1) and

01

+o(1) as n .

This is equivalent to the convergence of the product

n=1Bn() and its invertibility.For = 0 the matrix is reduced to

Bn=

1 00 1 2

n

+R(5)n ,

and the base of solutions changes accordingly:

10

+o(1) and 1

n2 01

+o(1)

as n ,

which follows from the discrete Levinson theorem [5, Theorem 2.2]. The existence of sucha base of solutions is in its turn equivalent to the convergence of the product

n=1Bn(0)

to a rank one matrix (since the second solution goes to zero as n ). Remark 3. The rank one matrix =

n=1Bn(0) defines the linear map

flimnun(0, f).Asymptotics of solutions of the equation un+1 = Bn()un given above demonstrate the

change of the system type. For non-zero the system is elliptic (i.e., its solutions have thesame rate of growth), while for = 0 the system is hyperbolic (i.e., there exists a base ofsolutions, which have uncomparable magnitudes; this yields the existence of a subordinate

solution).The following lemma states the uniform convergence of the product of matrices Bn()in the slow scale.


15/24


Lemma 6. Let{Rn()}n=1l1(U). Thenn>

y||

Bn()I asy+ uniformly inU\{0}.

Proof. The sequence V(1)n given by (31) has the following property:

(33)

kn

V(1)k ()

4n sin 2

for every n Nand U\{0}.This easily seen using the equality (32), elementary estimates

sin

1, cos 1

1 andLemma 2. It enables to define the sequence

Q(2)n :=k=n

V(1)k .

Then

Bn = I+ Q(2)n Q(2)n+1+R(5)n .

Following the ideas of [14, 5], we want to consider the Harris-Lutz transformation: B(1)n :=

I Q(2)n+11

Bn

I Q(2)n

. Ifn > c2|| with, say,

(34) c2:= 8 supU\{0}

||sin 2

,then it is easy to see that the estimateQ(2)n () < 12 holds yielding the invertibility of

I Q(2)n+1

. For such values ofnby a straightforward calculation one has:

(35) B(1)n = I Q(2)n+11I+Q(2)n Q(2)n+1I Q(2)n + I Q(2)n+11R(5)n I Q(2)n =I+

I Q(2)n+1

1Q

(2)n+1 Q(2)n

Q(2)n

=:V(2)n

+

I Q(2)n+11

R(5)n

I Q(2)n

=:R(6)n

.

Using the trivial boundsV(2)n < 2V(1)n Q(2)n, R(6)n < 3R(5)n, V(1)n 2n and theboundQ(2)n () 4

n|sin 2 | , one hasn> y

||

V(2)n () +R(6)n () n> y

||

162||ysin

2

+ 3

n> y

||

R(5)n ()0 as y+uniformly in

U

\{0}

. A rather rough argument repeating the scalar estimates yields:

n> y||

B(1)n ()

I exp

n> y||

V(2)n () +R(6)n () ,

hence the assertion of the lemma holds, ifBn is replaced by B(1)n . Then, coming back to

the product of matrices Bn():

(36)k> y

||

Bk() =

I limn

Q(2)n ()

k> y

||

B(1)k ()

I Q(2) y||+1()1

.

Due to the estimate (33), Q(2) y||() 0 as y + uniformly in U\{0}. Thereforethe convergence to the identity matrix in (36) is uniform. This completes the proof.


16/24


The next lemma completes Step I and proves the uniform boundedness of all partialproducts of matrices Bn(). This lemma is rather non-trivial and plays an importantrole in the rest of the proof. It is only due to the choice of the scaling factor in vn =

exp

n

1cos(r)

r dr

un that these products are uniformly bounded.

Lemma 7. Let{Rn()}n=1 l1(U). Then there exists a constant c3 such that for everyUand everyn

(37)

n

k=1

Bk()

< c3.Proof. Using the decomposition of the exponent (29) again one can rewrite the sequenceBn in the following form:

Bn() = 1 0

0 exp2n+1

n cos(r)r dr +V(3)n () +R

(7)n ()

with some{R(7)n ()}n=1l1(U) and

(38) V(3)n () :=

n

n+1n

cos(r)dr

1 00 1

+

n


=

n

cos(n)

n+1n

cos(r)dr

1 00 1

+

nsin(n)

0 11 0

.

We are going to perform the variation of parameters in the discrete equation

un+1 = 1 00 exp2n+1n

cos(r)r

dr +V(3)n () +R(7)n ()un,

considering it as a perturbation of the equation

un+1 =

1 00 exp

2

n+1n

cos(r)r

dr

un.This leads to the following:

(39) un(, f) = 1 00 exp2n1 cos(r)r dr f+

n1k=1

1 0

0 exp2n

k+1cos(r)

r dr

V(3)k () +R

(7)k ()

uk(, f).

Using the Gronwalls lemma and a simple estimate

(40) exp

2

yt

cos s

s ds

exp

2

32

2

cos s

s ds

32, if 0ty ,

one gets the following bound for the solution.

un(, f) 32exp32 n1k=1

V(3)k () +R(7)k () f.


17/24


In other terms,

(41)

nk=1

Bk()

33exp

33

nk=1

V(3)k () +R

(7)k ()

.

Furthermore, the identity (38) leads to the following estimate:V(3)n () 52|| for every nNand U,which can be easily obtained with the help of the equality (32) and explicit bounds cos 1

1, 1 sin ||

2. Thus, ifn c2|| (where c2 is defined by (34)), then

(42)n

k=1

V(3)k () +R

(7)() 5c2

2 +

k=1 R(7)()

including the case of= 0, n=.The estimate (41) grows to infinity with n, therefore the tail of the matrix product

ought to be considered separately. If = 0 and n > c2|| , one obtains from the equalityB

(1)n =I+ V

(2)n +R

(6)n , see (35):

nk= c2||+1

B(1)k ()

c2|| uniformly with respect to . Assertion of the lemma follows.

Step II. Rewriting the system un+1=Bn()un in the form of a Volterra integralequation in the slow scale. Consider the equation (39),

un(, f) =

1 0

0 exp2n

1cos(r)

r dr

f

+n1k=1

1 0

0 exp2n

k+1cos(r)

r dr

V(3)k () +R

(7)k ()

uk(, f).

which is equivalent to the systemun+1 = Bn()un. On this step we rewrite it in an integral

operator form. Fix an

U\{

0}

. Put againn = y|| and divide the sum in (39) intotwo sums, which contain terms R(7) and V(3) , respectively. Then write the second sum ofthe two as an integral in a new variable putting k =. Doing this we get for y ||,


18/24


since a piece-wise constant function appears in the second sum:

(44) u y||() =

1 0

0 exp

2

y||1

cos(r)r

dr

f

+

y||1k=1

1 00 exp

2 y||k+1 cos(r)r dr

R(7)k ()uk()+

y||1

1 00 exp

2 y||+1 cos(r)r dr

V(3) ()u()d.Scaling the variable of the integration = t|| we write this as a Volterra integral equation:

(45) z(y) =g(y) + y

0

K(y, t)z(t)dt.

Here we denote by z, g and Kthe following piecewise-constant (on intervals of length )functions:

z(y,,f) :=u y||(, f), ify ||,

(46) g(y,,f) :=

1 00 exp

2|| y|||| cos ss ds

f+

y||1

k=1 1 0

0 exp2 y||k+1 cos(r)r dr R(7)k ()uk(, f), ify ||,K(y,t,) :=

1 00 exp

2|| y||||( t||+1) cos ss ds

V(3) t||()|| , if|| t t > 0

(47) K(y,t,)

0 1exp

2yt

cos ss

ds

0

sin t

t as 0.

Additionally, for every y >0

(48) g(y, ) 1 00 0

f+

k=1

R(7)k (0)uk(0, f)

as 0.


19/24


This follows from the uniform bound given by Lemma 7 and from the property

{R(7)n ()}n=1l1(U) by Lebesgue dominated convergence theorem.Remark 4. It is very important that the limit in(48)coincides with the lim

nun(0, f). The

latter can be obtained from the following variant of discrete Levinson theorem[16, Lemma4.4, case (b)].

Proposition 2 ([16]). Suppose thatk=1

Rk|k| 0 andf C2 there exist two limitsh(y, f) := lim

0z(y,,f),

which satisfy the following integral equations:

(49) h(y, f) = limn

un(0, f) y0

0 1

exp2y

tcos ss

ds

0

sin t

t h(t, f)dt.

Proof. For every y0 > 0 and = 0 define the operatorKy0() in the Banach spaceL((0; y0),C2) by the rule

(50) Ky0() :u(y) y0

K(y,t,)u(t)dt, y(0; y0).Denote K(y,t, 0) := lim

0K(y,t,) and analogously define two operatorsKy0(0). We

consider only the case +0 here. The second case can be treated in the same way.First let us see that the operatorKy0() converges toKy0(+0) as +0 in the space

of bounded linear operators in L((0; y0),C2). It is enough to show that

(51) maxy[0;y0]

y0

K(y,t,) K(y,t, +0)dt0 as +0.Fix a > 0. Since both kernels are uniformly bounded in all variables, there exists ay1()< y0 such that for every positive

U

maxy[0;y1()]

y0

K(y,t,) K(y,t, +0)dt


20/24


For the same reason

(52) maxy[y1();y0]

y0

K(y,t,) K(y,t, +0)dt

= maxy[y1();y0] y

K(y,t,) K(y,t, +0)dt+O() as +0.Using the estimates 1cos

=O() and 1 sin

=O(2) as 0, one can write

K(y,t,) =

0 1exp

2y

( t+1)cos ss

ds

0

sin tt

+O() as +0for y and t such that 2 < y < y0 and < t < y , where O() is uniform with respectto these y and t. We remind that

K(y,t, +0) = 0 1

exp 2yt cos ss ds 0 sin t

t

.

The mapping (y; t) exp 2yt

cos sdss

is uniformly continuous on the compact set

y1() y y0, 0 t y (notice that it has a discontinuity at the point y =t = 0), the function t sin t

t is uniformly continuous in the interval [0; y0]. Therefore

maxt[;y] K(y,t,) K(y,t, +0) = o(1) as +0 uniformly in y [y1(); y0]. To-gether with (52) this means that for sufficiently small positive values ofthe maximum

maxy[y1();y0]

y0

K(y,t,) K(y,t, +0)dt 0 one has:g(y, ) lim

nun(0) as0.

We face an obstacle here: this convergence is only point-wise and is not in the norm ofL((0; y0),C2). The same is true for the convergence ofz(y, ), which we are going to prove.There is no sense in directly applying the inverse of the operator I Ky0(). The properobject to consider is the difference z(y, ) g(y, ), which converges inL((0; y0),C2). Tosee this let us rewrite the equation (45) in the following form:

(53) z() g() = (I Ky0())1Ky0()g()(sinceKy0() is a Volterra integral operator, (IKy0())1 exists). It suffices to prove thatKy0(+0)g()Ky0(+0) limnun(0) in L((0; y0),C2). By a direct estimate we have:

maxy[0;y0]

y0

K(y,t, +0)g(t, ) limn

un(0) dt32 y0

0

g(t, ) limn

un(0)dt.

The right-hand side tends to zero as +0 due to the point-wise convergence and theuniform boundedness ofg(t, ) by Lebesgue dominated convergence theorem. From (53)we obtain:

z() g()(I Ky0(+0))1Ky0(+0) limn

un(0) as+0 in L((0; y0),C2)

From the point-wise convergence ofg(y, ) it follows that for every y >0z(y, )

I+ (I Ky0(+0))1Ky0(+0)

limn

un(0)

(y) =:h+(y).


21/24


Consider the integral equation (45) again. Taking the limit as +0 turns it into theequation for h+, (49). This is possible due to the uniform boundedness of the solutionzand of the kernel K by Lebesgue dominated convergence theorem. This completes theproof.

Remark 5. The assertion of the last lemma includes the case when limnun(0, f) = 0.In this case h(y, f) 0. This happens only for one particular direction of the vectorf C2, because the rank of is one.Step IV. Convergence as y+ and changing the order of limits. It remains toshow that every solution of integral equations (49) has a limit at infinity and to see thatit is possible to change the order of limits in lim

y+lim0

u y||(, f).

Lemma 9. The following two limits exist: limy+

h(y, f). These limits are non-zero, iff

limn

un(0, f)= 0.

Proof. Consider the integral equation for h+:

h+(y) = limn

un(0, f) +

y0

0 1

exp2y

tcos ss

ds

0

sin t

t h+(t)dt.

Let as write it as a differential equation using the following factorization property of thematrix in the integral:

0 1exp

2yt

cos ss

ds

0

=

1 0

0 exp

2y

cos ss

ds

0 1

exp 2t cos ss ds 0 .Multiplying from the left by

1 0

0 exp

2y

cos ss

ds 1

and taking the derivative in y

one has:

h+(y)

y

0 sin ysin y 2cos y

h+(y) =

2cos y

y

0 00 1

limn

un(0, f).

By Lemma 5 the vector limn

un(0, f) C2 is either proportional to

10

or is equal to

zero and therefore

(54) h+(y) = y 0 sin ysin y 2cos y h+(y).If lim

nun(0, f)= 0, thenh+(y) is not identically zero. Every non-zero solution of (54) has a

non-zero limit asy+, which follows for example from Harris-Lutz results [14, Theorem3.1, p.85] (one should take there (y) =2cos y

y

0 00 1

,V(y) = sin y

y

0 11 0

,R(y) =

0). For the second limit (ofh(y, f)) the proof is absolutely analogous.

The following lemma is a combination of definitions and previous results: Lemmas 6, 8,9.

Lemma 10. The following two limits exist: lim0 limn un(, f) and equal tolim

y+h(y, f), respectively.


22/24


Proof. By definition of h, limy+

h(y, f) = limy+

lim0

u y||(, f). By Lemma 6,u y||(, f) converges uniformly with respect toU\{0}. Therefore there exist two limits,lim0

limy+

u y||(, f) which coincide with the limits limy+ lim0 u y||(, f), respectively.

Clearly limy+ u y||(, f) = limn un(, f) for= 0, which implies that lim0 limn un(, f) =

limy+

lim0

u y||(, f) = limy+ h(y, f). This completes the proof.

We are ready now to finish the proof of Theorem 1. Limits lim0

u y||(, f) exist byLemma 8, the equality lim

0limn

un(, f) = limy+

h(y, f) has sense and holds true by

Lemma 10. The map has the rank one by Lemma 5 (see also Remark 3).

5. Zeroes of the spectral density

In this section, we prove the main result of the paper by applying Theorem 1.

Theorem 2. Letq1 L1(R+), / Z2a , and() be the spectral density of the operatorL. Let the indexj be greater or equal to zero. If the solution(y, j,+) of (3) is not a

subordinate one, then there exist two non-zero limits

limj,+0

()

| j,+|2|c|

a|W{+(j,+),(j,+)}|

a

02+(t,j,+)e

2itdt

.

Analogously, if(y, j,) is not subordinate, then there exist two non-zero limits

limj,0

()

|

j,

|

2|c|a|W{+(j,+),(j,+)}|

a

02(t,j,)e

2itdt

.

Proof. We use the notation of Section 3: cr, instead ofj,, j,. This will yield bothclaims of the theorem. By (20), the solution(y, ) is related to the solution{v,n()}n=1of (19). In turn, the latter is related by the scaling

vn = exp

n1

cos(r)r

dr

un to the solution{un(, v,1())}n=1 of the system un+1 =Bn()un. We now apply Theorem to the sequence 1 to {un(, v,1())}n=1. Non-subordinacyof(cr) means that lim

nun(0, v,1(0))= 0. Due to continuity ofv1,() inU, Theorem 1

implies the existence of the following two limits

lim0

limn

un(, v,1())= 0.

Taking into account the scaling vn = expn1 cos(r)r drun this means that there existtwo non-zero limits lim

0limn

||v,n(). This in turn yields the assertion of the theoremdue to the Weyl-Titchmarsh type formula (21) and the property of the quasi-momentumthat its derivative is positive inside spectral bands.

6. Discussion

As mentioned in Introduction, our result is to be compared to the one of Hinton-Klaus-Shaw [15]. The difference in the classes of operators considered may seem significant: in[15] there is no periodic background and the non-summable part of the potential is an

infinite sum of Wigner-von Neumann type terms. Nevertheless, both problems can bewritten after a few suitable transformations in virtually the same form, see system (6) inour case and the formula prior to (2.2) in [15]. We are also able to consider finite or even


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infinite sum of Wigner-von Neumann type terms and to reduce the problem to the sameform by analogous transformations. The methods that we use also have common traitslike usage of the slow variable y =||n =||x/a or work with the limit integral Volterraequation like (27) which actually represents an equation of the same type as (2.75) in [15]but is written in a different, more explicit form. However, the approach that we develop inthe present paper to our mind has several essential distinctions compared to the approachof [15].

1. We use a discretization of the differential system and work with the discrete modelsystem of a simple form (19). In addition to bringing certain technical (inessential) dif-ficulties it makes our method more universal and enables one to consider a wider class ofproblems. As an example, we can analyze by a direct application of Theorem 1 the struc-ture of zeroes of the spectral density of the discrete Schrodinger operator with Wigner-vonNeumann potential. This is by far not the only possible example. Using the solution of themodel problem one can also consider differential Schrodinger operators with point interac-tions at integer points with strengths of interaction forming a sequence of the Wigner-von

Neumann form.2. We do not reduce the problem to a scalar one and use the analysis of vector equationsinstead. This makes the analysis simpler and more transparent. As a consequence ofthis, the form of the limit equation (27) is explicit and can be guessed from heuristicconsiderations (see (24)-(26)).

3. We avoid fine technically involved estimates of oscillatory integrals and in fact ourproof is divided into two parts. First (Step I), we prove rather rough a priori estimatesby dividing the positive half-line into subintervals R+=

0; c2||

c2|| ; + for sufficientlylargec2. Second (Steps II-IV), we deal with the slow scale and prove the convergence usingoperator techniques. This is the main distinction of our method which makes it sufficientlysimpler than the method of [15].

We expect that using the technique developed in the present paper one can treat a morecomplicated case of the Wigner-von Neumann perturbation of non-Coulomb type withslowly decaying power part. We plan to address this problem in the future.

Acknowledgements

Authors wish to express their gratitude to Dr. Roman Romanov for helpful remarksand fruitful discussions and to Dr. Alexander V. Kiselev for his help with the work onthe text and many useful remarks and comments. The work was supported by the grantRFBR-09-01-00515-a. The second author was also supported by the Chebyshev Laboratory(Department of Mathematics and Mechanics, Saint-Petersburg State University) under the

grant 11.G34.31.2006 of the Government of the Russian Federation.

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Department of Mathematical Physics, Institute of Physics, St. Petersburg University,

Ulianovskaia 1, St. Petergoff, St. Petersburg, Russia, 198904

E-mail address: [email protected]

Chebyshev Laboratory, Department of Mathematics and Mechanics, Saint-Petersburg

State University 14th Line, 29b, Saint-Petersburg, 199178 Russia

E-mail address: sergey [email protected]

zeros of spectral density of von neumann schrodinger

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