A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

Hλu = (H0 + λWn) un = Eun, n ∈ [0,∞) , (1) in the Hilbert space l2(Z) of all square summable sequences {xn}∞1 such that ∑1 |xn|2 < ∞. Here, H0 = −Δ + Vn, the Laplacian −Δ is a second-order finite difference operator given by (−Δu)(n) := −un+1 + 2un − un−1. The operator H0 describes a well-understood system with Vn being the background periodic or quasi-periodic potential. The parameter λ, called the coupling constant, is real and positive and E is a complex parameter. The sequences {Vn}∞0 are real and are such that Va+n = Va, for some a ∈ Z+.The perturbing termWn satisfies the scattering condition


Introduction
We consider the Schrödinger difference equation in the Hilbert space  2 (Z) of all square summable sequences Here,  0 = −Δ +   , the Laplacian −Δ is a second-order finite difference operator given by (−Δ)() := − +1 + 2  −  −1 .The operator  0 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 {  } ∞ 0 are real and are such that  + =   , for some  ∈ Z + .The perturbing term   satisfies the scattering condition The domain    of   consists of all  ∈  2 (Z) such that    ∈  2 (Z).The operators   and  0 are known to be selfadjoint [1,2].Operators of the form   =  0 + , 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 Al 2 O 3 crystal, which has a large gap between the first and the second bands: it is transparent and colourless.By replacing the Al + -ion by Cr 3+ , 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][7][8][9].
The spectrum ( 0 ) of the self-adjoint operator  0 is real and consists of two absolutely continuous bands separated by a gap.By a spectral gap of  0 , we mean an interval (, ) such that  < , ,  ∈ ( 0 ), and (, ) ∩ ( 0 ) ̸ = 0. We choose our   's such that the gaps are empty when  = 0.For  > 0, the spectrum of   is made up of the continuous part which coincides with that of ( 0 ) together with at most a finite number of eigenvalues inside the gap of ( 0 ) 2 Advances in Mathematical Physics [3,[10][11][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  → 0 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 [0,4].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  = 0, 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][15][16][17][18].We call  0 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  ↑  0 or  ↓  0 , respectively, or both.In particular, the case where () ↓  as  ↓  0 corresponds to the situation where an eigenvalue appears at  as  ↓  0 + .For more results in connection with the Schrödinger case, see, for example, [3].
The purpose of this work is to (1) study the analytic behaviour of the eigenvalue () near the coupling constant threshold and (2) study the asymptotic behaviour of the discrete spectrum in the gaps of the spectrum of  0 as the coupling constant grows to infinity.Our main tool will be the Birman-Schwinger operator and the Titchmarsh-Weyl function theory.

The Birman-Schwinger Principle
The spectrum ( 0 ) of  0 is real and consists of two absolutely continuous bands covering the closed intervals Since   is assumed to be relatively  0 -compact, we also have that  ess ( 0 +   ) =  ess ( 0 ).
If we let () ∈ ( 1 ,  2 ), then the Birman-Schwinger kernel is defined by [19,20] The Birman-Schwinger principle implies the following.Let  0 be a self-adjoint operator and  ∈ ( 0 ), and suppose that   ≥ 0 is a bounded operator with   ( 0 −) −1 compact.Then the Birman-Schwinger kernel   is compact and the following are equivalent: (1)  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   ≤ 0 (  ≥ 0), and then the nonzero eigenvalues of   are strictly monotone increasing (or decreasing) respectively.Proposition 2. There exists a  ∈ R such that  0 +   has an eigenvalue in ( 1 ,  2 ).
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].6) suppose that  = 0 is not a coupling constant threshold and dim  ( 1 , 2 ) ( 0 +   ) < ∞ for all  > 0. Then   2 is compact, (7)   2 exists and is not compact iff  = 0 is not a coupling constant threshold and there exists The threshold behaviour is described in the following lemma.

The 𝑚(𝐸)-Function Theory
Let   (, ) and   (, ) be the two linearly independent solutions of (1) satisfying the following initial conditions for all  ∈ C: and then there exists a solution   (, ) =   (, ) + (, )  (, ), which is in  2 (0, ∞).The function (, ) is defined by The general proof for the existence of the limit in (5) may be found in [1,21].The ()-function is analytic for Im() ̸ = 0, and it has a nonreal limit as Im() → 0 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 halfbound states or otherwise.() =  1 is a half-bound state (HBS) provided that {  ()} is abounded sequence but not in  2 (0, ∞) 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 system (6) with ∑ ∞ 0 |  | < ∞.Imposing the boundary conditions  0 +  1 = 0, the spectrum of ( 6) is [0, 4], with a finite number of eigenvalues in the intervals (−∞, 0) ∪ (4, ∞).At an eigenvalue  0 ∈ (−∞, 0) ∪ (4, ∞), we have that At the points  = 0, 4 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.

The Bound States of 𝐻 0 + 𝜆𝑊 𝑛
We consider the following system: where  + =   , for some  ∈ Z, and ∑ ||  < ∞.Let   ,   be the solutions of  0   =   satisfying the following initial conditions:  0 = 1,  1 = 0, and and this defines the fundamental matrix for ( 0 − ) = 0, and moreover (+) = ()() and det () = det (0) = 1.The eigenvalues of () are The gaps and bounds are given by |()| > 2 and |()| > 2, respectively.Let us just look at the specific situation where  is near  1 .We set and notice that we can view  as an even function of  in the vicinity of  1 ; that is, for  small so that  goes around  1 , √  2 − 4 → − √  2 − 4, and  + → − − , that is,  → −.Denoting the eigenfunctions corresponding to  ± by  ± where provided   ̸ = 0 (if   = 0, we shift the origin), we now define the periodic vector functions with q± () = q± ( + ).Let and it follows that p± ( + ) = p± ().Let These are the exponential solutions which behave in the following way: As for the Wronskian W = (Ψ + , Ψ − ), we have We have that   < 0 (at the right endpoint of a gap where () > 2).This follows from general principles below since when   < 0, the eigenvalues in the gap must appear at the right endpoint if  increases.

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.
It follows directly in this way: if we write Ψ − =  + , where  = ( − −   )/  , then () is the Titchmarsh-Weyl function.In the Schrödinger case Im () is positive for Im  > 0. Let us see what it is here: from