Abstract

The asymptotic and threshold behaviour of the eigenvalues of a perturbed difference operator inside a spectral gap is investigated. In particular, applications of the Titchmarsh-Weyl -function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator inside the spectral gap of in the limits and

1. Introduction

We consider the Schrödinger difference equationin the Hilbert space of all square summable sequences such that Here, , the Laplacian is a second-order finite difference operator given by The operator describes a well-understood system with being the background periodic or quasi-periodic potential. The parameter , called the coupling constant, is real and positive and is a complex parameter. The sequences are real and are such that , for some The perturbing term satisfies the scattering conditionThe domain of consists of all such that The operators and are known to be self-adjoint [1, 2].

Operators of the form , either in the difference or in the continuum form, occur frequently in quantum mechanics as mathematical models of Schrödinger type that models semiconductors with impurities [1, 3]. The impurity levels in solids reduce the width of the gap where they are situated, and this feature has important consequences from the point of view of conductivity properties of the solids. Furthermore, such impurity levels lead to a selective absorption of certain photon energies which is an important element in the theory of crystal colours. Perhaps the most appealing example in nature is the crystal, which has a large gap between the first and the second bands: it is transparent and colourless. By replacing the -ion by , one obtains a familiar complementary colour red, and this is due to the fact that the impurity levels lead to the absorption of green light [4, 5]. Recent and relevant results on the application of the solutions of Schrödinger equations with physical potential models in the field of thermal physics, for example, can be found in the work of Jia et al. [6–9].

The spectrum of the self-adjoint operator is real and consists of two absolutely continuous bands separated by a gap. By a spectral gap of , we mean an interval such that , and We choose our ’s such that the gaps are empty when For , the spectrum of is made up of the continuous part which coincides with that of together with at most a finite number of eigenvalues inside the gap of [3, 10–12]. Of interest to this work are the conditions on guaranteeing the existence of such eigenvalues and their asymptotic behaviour as varies.

The case where is not periodic but decays fast enough to zero as was studied in [13], and a formula of the type known as Levinson’s theorem was derived which counts the number of eigenvalues or bound states outside the continuous band . We would like to obtain the corresponding results when is periodic and determine how those eigenvalues vary as the coupling constant becomes large and small. Some results do exist for the Schrödinger case; however, in the discrete case this has not been looked at (see, e.g., [3]).

Since there are no eigenvalues when , by general results of perturbation theory, eigenvalues of can appear in a gap only by emerging from one of its end points as is varied. Likewise, eigenvalues can disappear from the gap only by converging to an end point [14–18]. We call a coupling constant threshold of the family of the operators at the gap end points and if there exists an eigenvalue branch of such that or , as either or , respectively, or both. In particular, the case where as corresponds to the situation where an eigenvalue appears at as For more results in connection with the Schrödinger case, see, for example, [3].

The purpose of this work is to study the analytic behaviour of the eigenvalue near the coupling constant threshold and study the asymptotic behaviour of the discrete spectrum in the gaps of the spectrum of as the coupling constant grows to infinity. Our main tool will be the Birman-Schwinger operator and the Titchmarsh-Weyl -function theory.

2. The Birman-Schwinger Principle

The spectrum of is real and consists of two absolutely continuous bands covering the closed intervals , where . Since is assumed to be relatively -compact, we also have that

If we let , then the Birman-Schwinger kernel is defined by [19, 20]The Birman-Schwinger principle implies the following.

Let be a self-adjoint operator and , and suppose that is a bounded operator with compact. Then the Birman-Schwinger kernel is compact and the following are equivalent:(1) is an eigenvalue of with multiplicity .(2) is an eigenvalue of with multiplicity .

The following results are well known, we state them without proof, and the proofs for the continuous case can be found in [3], for example.

Proposition 1. Suppose that , and then the nonzero eigenvalues of are strictly monotone increasing (or decreasing) respectively.

Proposition 2. There exists a such that has an eigenvalue in

The following lemma summarizes all the information concerning the operator and the behaviour of the eigenvalues inside the spectral gap, and its proof can be found in [3].

Lemma 3. Let Then(1)if is a coupling constant threshold of then is unbounded as ,(2)if stays bounded as , then exists,(3) is compact iff in norm,(4)if for some then for all ,(5) is compact iff for all ,(6)suppose that is not a coupling constant threshold and for all . Then is compact,(7) exists and is not compact iff is not a coupling constant threshold and there exists such that for and for .

The threshold behaviour is described in the following lemma.

Lemma 4. Suppose that and suppose is not a threshold. Then is not an eigenvalue of iff all eigenvalues which are absorbed at obey

3. The -Function Theory

Let and be the two linearly independent solutions of (1) satisfying the following initial conditions for all : and then there exists a solution , which is in The function is defined byThe general proof for the existence of the limit in (5) may be found in [1, 21]. The -function is analytic for , and it has a nonreal limit as in the bands and is real in the gaps except for the poles of at the eigenvalues of Of interest is the behaviour of at the end points of the gaps, if the endpoints are either half-bound states or otherwise. is a half-bound state (HBS) provided that is abounded sequence but not in and a non-half-bound state (non-HBS) otherwise. A similar definition may be given at the other end point of the spectral gap. In [13], there are the results of a version of Levinson’s Theorem for the systemwith

Imposing the boundary conditions , the spectrum of (6) is , with a finite number of eigenvalues in the intervals At an eigenvalue , we have that At the points it turns out that It is natural to ask whether the theory of [13] carries over to the periodic version (1) in the presents of a gap and possible half-bound states at the ends of each gap.

4. The Bound States of +

We consider the following system:where , for some , and Let be the solutions of satisfying the following initial conditions: , and

Letand this defines the fundamental matrix for , and moreover and The eigenvalues of are The gaps and bounds are given by and , respectively. Let us just look at the specific situation where is near We set and notice that we can view as an even function of in the vicinity of ; that is, for smallso that goes around , , and , that is, Denoting the eigenfunctions corresponding to by whereprovided (if , we shift the origin), we now define the periodic vector functions with Letand it follows that LetThese are the exponential solutions which behave in the following way:As for the Wronskian , we haveWe have that (at the right endpoint of a gap where ). This follows from general principles below since when , the eigenvalues in the gap must appear at the right endpoint if increases. It follows directly in this way: if we write , where , then is the Titchmarsh-Weyl function. In the Schrödinger case is positive for Let us see what it is here: from summing from 1 to giveswhere so that Im when If is close to in the upper half plane, then and hence

Now let us consider the Birman-Schwinger kernel:and, assuming , has eigenvalue 1 iff The resolvent is given bywith Using (17), we get (using as a variable)where is the first component of . is analytic in with We note thatWe aim to determine the limit as of There is a diverging rank one piece which we can isolate.whereThe first term gives a rank one contribution to and it diverges as , with The other terms when sandwiched between go to finite limits in Hilbert Schmidt norm. The rank one piece gives rise to a bound state near in the weak coupling limit (if , even if changes sign). If , then the eigenvalue gets absorbed at as

5. Conclusion

By using the standard results derived for the continuous Schrödinger operator case, some insights into the asymptotic behaviour of eigenvalues of the difference Schrödinger operator in the limit of the large and small coupling constant have been derived. This forms the basis for future work on the investigation of the asymptotic behaviour of the number of the eigenvalues inside the spectral gap for a large coupling constant which are not covered in this note.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Martin Klaus (Virginia Tech) for several useful discussions at the time when this paper was conceived of. Gift Muchatibaya and Josiah Mushanyu would like to acknowledge the support of the University of Zimbabwe when this project was underway.