Spectral analysis of the half-line Kronig-Penney model with Wigner-von Neumann perturbations
The spectrum of the self-adjoint Schrödinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points “instable” embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains discrete.
Key words and phrases:Wigner-von Neumann potentials, point interactions, Kronig-Penney model, embedded eigenvalues, subordinacy theory, discrete linear systems, asymptotic integration, compact perturbations.
2010 Mathematics Subject Classification:Primary 34B24; Secondary 47E05, 34L40
In the classical paper [vNW29] von Neumann and Wigner studied the one-dimensional Schrödinger operator with the potential of the form and discovered that such an operator may have an eigenvalue at the point of the continuous spectrum . Since then such potentials permanently attracted interest [Al72, Be91, Be94, HKS91, L10, Ma73, N07, RS78]. See also recent developments in this context on the counterpart problem for Jacobi matrices [JS10, S12] and for the case of periodic differential Schrödinger operators with Wigner-von Neumann perturbations [KN07, KS13, LO13, NS12].
In parallel to the progress in the investigation of Wigner-von Neumann potentials considerable interest was attracted by Schrödinger operators with point -interactions [GK85, GO10, Ko89, KM10, L11, M95, SCS94] and also with more general distributional potentials [EGNT13, ET13, R05, SS99, SS03] (our list of references is of course by no means complete).
So let the discrete set with elements enumerated in the increasing order be such that
and the real-valued sequence be arbitrary. We deal with the self-adjoint Schrödinger operator in with point interactions of strengths supported on . This operator corresponds to the formal differential expression
and to the boundary condition at the origin
See Section 2.1 for the mathematically rigorous definition of such operators. As a special case the Kronig-Penney model corresponds to the self-adjoint operator as above with some and . It describes the behaviour of a free non-relativistic charged quantum particle interacting with the lattice . The constant characterizes the strength of interaction between the particle and each interaction center in the lattice. This interaction can be repulsive (), attractive () or absent (). The spectrum of the operator has a band-gap structure: it consists of infinitely many bands of the purely absolutely continuous spectrum and outside these bands the spectrum of is discrete, cf. [AGHH05, Chapter III.2]. The operator was first investigated in the classical paper [KP31] by Kronig and Penney.
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 the constants and a real-valued sequence be such that
Model I: Wigner-von Neumann amplitude perturbation. We add a discrete Wigner-von Neumann potential to the constant sequence of interaction strengths. Namely, we consider the discrete set and the sequence of interaction strengths given by
We study the self-adjoint operator which reflects an amplitude perturbation of the Kronig-Penney model.
Model II: Wigner-von Neumann positional perturbation. We change the distances between interaction centers in a “Wigner-von Neumann” way, i.e., we add a sequence of the form of Wigner-von Neumann potential to the coordinates of interaction centers leaving the strengths constant. Let the discrete set and the sequence of strengths be
We study the operator which reflects a positional perturbation of the Kronig-Penney model and describes properties of one-dimensional crystals with global defects.
The essential spectrum of the operator has a band-gap structure similar to the case of Schrödinger operator with regular periodic potential:
The locations of boundary points of the spectral bands are determined by the parameters and . Namely, the values are the -th roots of the corresponding Kronig-Penney equations
and the value is defined via extension by continuity to the point of the above function. For the details the reader is referred to the monograph [AGHH05, Chapter III.2].
In the present paper we show that the absolutely continuous spectra of the operators and coincide with the absolutely continuous spectrum of the non-perturbed operator . However the spectrum in bands may not remain purely absolutely continuous. Namely, at certain points which are called critical embedded eigenvalues may appear. In each band there are two such points. The critical points in the -th spectral band are the -th roots of the equations
The illustration is given in Figure 1.
For the considered operators we give exact conditions which ensure that a given critical point is indeed an embedded eigenvalue for some . For a given point this can occur only for one value of . We calculate the asymptotics of generalized eigenfunctions for all values of the spectral parameter , except the endpoints of the bands. The possibility of the appearance of an embedded eigenvalue at certain critical point depends on the rate of the decay of the subordinate generalized eigenfunction. It turns out that for Model I due to the decay rate of generalized eigenfunctions embedded eigenvalues can appear only in the low lying bands of the absolutely continuous spectrum, whereas for Model II one can create arbitrarily large embedded eigenvalues varying . We also show that the spectrum in gaps remains discrete.
Our results are close to the results for one-dimensional Schrödinger operator with the Wigner-von Neumann potential and a periodic background potential. Such operators were considered recently in [KN07, KS13, LO13, NS12].
To study spectra in bands we make a discretization of the spectral equations and further we perform an asymptotic integration of the obtained discrete linear system using 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 use compact perturbation argument.
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.
The body of the paper contains two parts: the preliminary part — which consists 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 Schrödinger operators with -interactions (Section 2.1), show how to reduce the spectral equations for these operators to discrete linear systems in (Section 2.2), provide a formulation of the analogue of the subordinacy theory for the considered operators (Section 2.3). Further we formulate few results from asymptotic integration theory for discrete linear systems (Section 2.4). In the main part, in Section 3.1 we study a special class of discrete linear systems in and find asymptotics of solutions of these systems. After certain technical preliminary calculations in Section 3.2, we proceed to Section 3.3, where we obtain asymptotics of generalized eigenfunctions for Schrödinger operators with point interactions subject to Model I and Model II. Further we pass to the conclusions about the spectra in bands putting an emphasis on critical points. In Section 3.4 we prove compactness of resolvent differences of two Schrödinger operators with point interactions under certain assumptions on interaction strengths and positions of interactions. This result is used to show that the spectrum in gaps of the Schrödinger operators with point interactions subject to Models I and II remains discrete.
By small letters with integer subindices, e.g. , we denote sequences of complex numbers. By small letters with integer subindices and arrows above, e.g. , we denote sequences of -vectors. By capital letters with integer indices, e.g. , we denote sequences of matrices with complex entries. We use notations , and for spaces of summable (), square-summable () and bounded () sequences of complex values, complex two-dimensional vectors and complex matrices, respectively. For a self-adjoint operator we denote its pure point, absolutely continuous, singular continuous, essential and discrete spectra by , , , and , respectively. Throughout the paper for we choose by default the branch of the square root so that if the opposite is not said.
2.1. Definition of operators with point interactions
In this section we give a rigorous definition of operators with -interactions, see, e.g., [GK85, Ko89]. Let be a sequence of real numbers and let be a discrete set on ordered as . Assume that the set satisfies
Denote also . In order to define the operator corresponding to the formal expression111In (2.2) we denote usual -distribution supported on by .
and the boundary condition
consider the following set of functions:
and let the operator be defined in by its action
on the domain
According to [GK85, Theorem 3.1] the operator is self-adjoint.
The spectral equation is understood as the equation for . The latter is equivalent to the following system:
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 (2.3) to a discrete linear system. Let the discrete set and the sequence of strengths be as in the previous section. Fix and set . To make our formulas more compact we introduce the following notations
Note that for the value is purely imaginary, and in this case we use identities and .
For a solution of (2.3) we introduce the sequence
Inversely, solutions of the eigenfunction equation on each of the intervals can be recovered from their values at the endpoints and :
The reader may confer with [E97], where a more general case of a quantum graph is considered.
Instead of working with recurrence relation (2.4) we will consider a discrete linear system in . Define
Observe that (2.4) can be rewritten as
The above recurrence relation is then equivalent to
The coefficient matrix of this system is called the transfer-matrix.
The case requires separate consideration. In this special case , but, in all the formulas, expressions of the form should be substituted by , its limit as and at the same time solution of the equation which is zero at the point and has the derivative equal to one there, cf. [E97]. This gives
Instead of (2.7) one gets
The subordinacy theory as suggested in [GP87] by D. Gilbert and D. Pearson produced a strong influence on the spectral theory of one-dimensional Schrödinger operators. Later on the subordinacy theory was translated to difference equations [KP92]. For Schrödinger operators with -interactions there exists a modification of the subordinacy theory, see, e.g., [SCS94] which relates the spectral properties of the operator with the asymptotic behavior of the solutions of the spectral equation (2.3). Analogously to the classical definition of the subordinacy [GP87] we say that a solution of the equation is subordinate if and only if for any other solution of the same equation not proportional to the following limiting property holds:
We will use the following propositions to locate the absolutely continuous spectrum.
Let the discrete set on be ordered as . Assume that satisfies conditions and . Let be a sequence of real numbers. Let and set . In the case assume also that holds. If every solution of the equation (see (2.3)) is bounded, then for such there exists no subordinate solution.
Let be an arbitrary solution of . Set for . If , then differentiating (2.5) one gets in the case
There exists such that . Since the sequence is bounded, one has that is also bounded for Obviously, for , is bounded too, since it is piecewise continuous with finite jumps at the points of discontinuity. If , then one gets
Boundedness of and the above formula imply boundedness of in the case as well.
Let be any other solution of , which is linearly independent with . It follows that there exists a constant such that
The Wronskian of the solutions and is independent of :
This is easy to check: it is constant on every interval and at the points one has
and therefore there exist constants and such that
Now we apply the trick used in the proof of [S92, Lemma 4]. We consider for an arbitrary the interval . Since the function is continuously differentiable on , the formula
Next we show that there exists a point such that . Let us suppose that such a point does not exist, i.e. for all . We get from (2.12) that for every . In particular is sign-definite in , so using (2.13) and we get a contradiction
Thus the point with required properties exists.
Since , for every such that one has . We have shown that every interval contains a subinterval of length , where . Therefore
On the other hand,
Summing up, for every solution the integral has two-sided linear estimate. Thus no subordinate solution exists. ∎
2.4. Benzaid-Lutz theorems for discrete linear systems in
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 systems to the case of discrete linear systems. The major advantage of these methods is that they allow to reduce under certain assumptions the asymptotic integration of some general discrete linear systems to the asymptotic integration of diagonal discrete linear systems. For our applications it is sufficient to formulate Benzaid-Lutz theorems only for discrete linear systems in . The first lemma of this subsection is a direct consequence of [BL87, Theorem 3.3].
Let be such that and let . If the coefficient matrix of the discrete linear system
is non-degenerate for every , then this system has a basis of solutions with the following asymptotics:
where by and we denote the diagonal entries of the matrices , and the factors and should be replaced by 1 for those values of the index for which they vanish (only a finite number).
The following lemma is a simplification of [BL87, Theorem 3.2].
Let and be such that and that the sum is (conditionally) convergent with
If for every
then the discrete linear system
has a basis of solutions with the following asymptotics:
where the factor should be replaced by 1 for those values of the index for which it vanishes (only a finite number).
3. Spectral and asymptotic analysis
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 (2.6) corresponding to and as in Models I and II described in the introduction.
Let the parameters , , , and satisfy the conditions
For further purposes we define
with any choice of the branch of the square root (although we specify it explicitly below in the subcase ). Define further
The following lemma has technical nature and helps to simplify the analysis of cases (Model I and Model II).
is non-degenerate for every , then this system has a basis of solutions with the following asymptotics.
If , then
If , then
If , then
Proof of Lemma 3.1.
Since , the constant term in the coefficient matrix in (3.5) can be diagonalized as follows
In view of the identity
transforms the system (3.5) on into the system on given below
with some .