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

Authors: Gift Muchatibaya; Josiah Mushanyu
Journal: Advances in Mathematical Physics
 (2018)

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## 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-Weylm-function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator H0+λWn inside the spectral gap of H0 in the limits λ↑∞ and λ↓0.


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## Body

## 1. Introduction

We consider the Schrödinger difference equation(1)Hλu=H0+λWnun=Eun,n∈0,∞,in the Hilbert space l2Z of all square summable sequences {xn}1∞ such that ∑1∞‍xn2<∞. 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 Vn0∞ are real and are such that Va+n=Va, for some a∈Z+. The perturbing term Wn satisfies the scattering condition(2)∑n=0∞nWn<0.The domain DHλ of Hλ consists of all x∈l2(Z) such that Hλx∈l2(Z). The operators Hλ and H0 are known to be self-adjoint [1, 2].Operators of the formHλ=H0+λW, 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 Al2O3 crystal, which has a large gap between the first and the second bands: it is transparent and colourless. By replacing the Al+-ion by Cr3+, 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 spectrumρ(H0) of the self-adjoint operator H0 is real and consists of two absolutely continuous bands separated by a gap. By a spectral gap of H0, we mean an interval (a,b) such that a<b,a,b∈ρ(H0), and (a,b)∩ρ(H0)≠∅. We choose our Vn’s such that the gaps are empty when λ=0. For λ>0, the spectrum of Hλ is made up of the continuous part which coincides with that of ρ(H0) together with at most a finite number of eigenvalues inside the gap of ρ(H0) [3, 10–12]. Of interest to this work are the conditions on Wn guaranteeing the existence of such eigenvalues and their asymptotic behaviour as λ varies.The case whereVn is not periodic but decays fast enough to zero as n→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 Vn 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 whenλ=0, by general results of perturbation theory, eigenvalues of Hλ can appear in a gap only by emerging from one of its end points as λ is varied. Likewise, eigenvalues can disappear from the gap only by converging to an end point [14–18]. We call λ0 acoupling constant threshold of the family of the operators Hλ at the gap end points a and b if there exists an eigenvalue branch E(λ) of Hλ such that E(λ)↓a or E(λ)↑b, as either λ↑λ0 or λ↓λ0, respectively, or both. In particular, the case where E(λ)↓a as λ↓λ0 corresponds to the situation where an eigenvalue appears at a as λ↓λ0+ϵ. 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 E(λ) near the coupling constant threshold and (2) study the asymptotic behaviour of the discrete spectrum in the gaps of the spectrum of H0 as the coupling constant grows to infinity. Our main tool will be the Birman-Schwinger operator and the Titchmarsh-Weyl m-function theory.


## 2. The Birman-Schwinger Principle

The spectrumρ(H0) of H0 is real and consists of two absolutely continuous bands covering the closed intervals [E0,E1]∪[E2,E3], where 0<E0<E1<E2<E3. Since Wn is assumed to be relatively H0-compact, we also have that ρess(H0+λWn)=ρess(H0).If we letE(λ)∈(E1,E2), then theBirman-Schwinger kernel is defined by [19, 20](3)KE=Wn1/2H0-EWn1/2.The Birman-Schwinger principle implies the following.LetH0 be a self-adjoint operator and E∈ρ(H0), and suppose that Wn≥0 is a bounded operator with Wn(H0-E)-1 compact. Then the Birman-Schwinger kernel KE is compact and the following are equivalent:(1)
E is an eigenvalue of Hλ with multiplicity m.(2)
λ-1 is an eigenvalue of KE with multiplicity m.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 thatWn≤0(Wn≥0), and then the nonzero eigenvalues of KE are strictly monotone increasing (or decreasing) respectively.Proposition 2.
There exists aλ∈R such that H0+λWn has an eigenvalue in (E1,E2).The following lemma summarizes all the information concerning the operatorKE and the behaviour of the eigenvalues inside the spectral gap, and its proof can be found in [3].Lemma 3.
LetWn≤0(Wn≠0). Then(1)
ifλ=0 is a coupling constant threshold of H0+λWn then KE is unbounded as E↑E2,(2)
ifKE stays bounded as E↑E2, then limE↑E2KE=KE2 exists,(3)
KE2 is compact iff KE→KE2 in norm,(4)
ifdimE(E1,E2)(H0+λWn)=∞ for some λ=λ1>0 then dimE(E1,E2)(H0+λWn)=∞ for all λ>λ1,(5)
KE2 is compact iff dimE(E1,E2)(H0+λWn)<∞ for all λ>0,(6)
suppose thatλ=0 is not a coupling constant threshold and dimE(E1,E2)(H0+λWn)<∞ for all λ>0. Then KE2 is compact,(7)
KE2 exists and is not compact iff λ=0 is not a coupling constant threshold and there exists λ∗>0 such that dimE(E1,E2)(H0+λWn)=∞ for λ>λ∗ and dimE(E1,E2)(H0+λWn)<∞ for λ<λ∗.The threshold behaviour is described in the following lemma.Lemma 4.
Suppose thatWn≤0(Wn≠0) and suppose λ=0 is not a threshold. Then E2 is not an eigenvalue of H0 iff all eigenvalues Ei(λ) which are absorbed at λ=0 obey (Ei(λ)-E2)λ-1→0.


## 3. Them(E)-Function Theory

Letθn(E,λ) and ϕn(E,λ) be the two linearly independent solutions of (1) satisfying the following initial conditions for all E∈C:(4)θ0E,λ=ϕ1E,λ=1,θ1E,λ=ϕ0E,λ=1, and then there exists a solution ψn(E,λ)=θn(E,λ)+m(E,λ)ϕn(E,λ), which is in l2(0,∞). The function m(E,λ) is defined by(5)mE,λ=-limn→∞θnE,λϕnE,λ.The general proof for the existence of the limit in (5) may be found in [1, 21]. The m(E)-function is analytic for Im(E)≠0, and it has a nonreal limit as Im(E)→0 in the bands and is real in the gaps except for the poles of m(E) at the eigenvalues of Hλ. Of interest is the behaviour of m(E) at the end points of the gaps, if the endpoints are either half-bound states or otherwise. E(λ)=E1 is a half-bound state (HBS) provided that {ϕn(E)} is abounded sequence but not in l2(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)-un+1+2un-un-1+Vnun=Eun,n∈0,∞,with ∑0∞‍nVn<∞.Imposing the boundary conditionsu0+αu1=0, the spectrum of (6) is [0,4], with a finite number of eigenvalues in the intervals (-∞,0)∪(4,∞). At an eigenvalue E0∈(-∞,0)∪(4,∞), we have that(7)E-E0mE⟶k1≠0asE⟶E0. At the points E=0,4 it turns out that(8)E-Ei1/2mE⟶k2≠0,ifEi=0or4isHBS0,else. 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 ofH0 + λWn

We consider the following system:(9)H0+λWnun=-un+1+2un-un-1+Vn+λWnun=Eun,un∈l2Z,where Vn+a=Vn, for some a∈Z, and ∑‍nWn<∞. Let θn,ϕn be the solutions of H0un=Eun satisfying the following initial conditions: θ0=1,θ1=0, and ϕ0=0,ϕ1=1Let(10)ψn=θnϕnθn+1ϕn+1,i.eψ0=1001,and this defines the fundamental matrix for (H0-E)u=0, and moreover ψ(n+a)=ψ(n)ψ(a) and detψ(a)=detψ(0)=1. The eigenvalues of ψ(a) are(11)μ±=D±D2-42,D=θa+ϕa+1. The gaps and bounds are given by DE>2 and DE>2, respectively. Let us just look at the specific situation where E is near E1. We set(12)α=ln⁡μ+=ln⁡D+D2-42, and notice that we can view E as an even function of α in the vicinity of E1; that is, for α small(13)D2-4=2+cE-E1+⋯2-4=2cE1-E1+powersofE-E1,so that E goes around E1, D2-4→-D2-4, and μ+→-μ-, that is, α→-α. Denoting the eigenfunctions corresponding to μ± by q± where(14)q±=1,μ±-θaϕa,provided ϕa≠0 (if ϕa=0, we shift the origin), we now define the periodic vector functions(15)q±~n=ψnq±,0≤n≤a-1, with q±~(n)=q±~(n+a). Let(16)fn=0,0≤n≤a-1fn+a=fn+1,elsewhere,p±~n=e±fn∓n/aαq±~n,and it follows that p±~(n+a)=p±~(n). Let(17)Ψ±=ϕnq±n=e±n/aαp±~n.These are the exponential solutions which behave in the following way:(18)Ψ±⟶0asn⟶∓∞,As for the Wronskian W~=W(Ψ+,Ψ-), we have(19)W~=μ--μ+ϕa=-D2-4ϕa.We have that ϕa<0 (at the right endpoint of a gap where D(E)>2). This follows from general principles below since when Wn<0, the eigenvalues in the gap must appear at the right endpoint if E increases. It follows directly in this way: if we write Ψ-=θ+mϕ, where m=(μ--θa)/ϕa, then m(E) is the Titchmarsh-Weyl function. In the Schrödinger case Imm(E) is positive for ImE>0. Let us see what it is here: from(20)un+1-2un+un-1+Vnun=Eun,u¯n+1-2u¯n+u¯n-1+Vnu¯n=E¯u¯n, summing from 1 to ∞ gives(21)u0u¯1-u¯0u1=E-E¯∑1∞unu¯n,where u0u¯1-u¯0u1=m¯(E)-m(E) so that Im m(E)<0 when Im(E)>0. If E is close to E1 in the upper half plane, then Imμ->0 and hence ψa<0.Now let us consider the Birman-Schwinger kernel:(22)KE=Wn1/2H0-EWn1/2,and, assuming Wn≤0, KE has eigenvalue 1 iff E∈ρ(H). The resolvent (H0-E)-1 is given by(23)H0-E-1un=1W¯Ψn-∑-∞n-1Ψm+um+1W¯Ψn+∑n∞Ψm-um=∑Rnmum,with(24)Rmn=1W¯Ψn-Ψm+χn≥m+1+1W¯Ψn+Ψm-χm≥n. Using (17), we get (using α as a variable)(25)Rmn=1W¯e-αn-mp-n,αp+m,αχn≥m+1+1W¯e-αn-mp+n,αp-m,αχm≥n,where p±(n,α) is the first component of p±~(n). p+(n,α) is analytic in α with p+(n,α)=p+(n,-α). We note that(26)W~=-2sinh⁡αψa>0,E<E1.We aim to determine the limit as α→0 of Rmn. There is a diverging rank one piece which we can isolate.(27)Rmn=1W¯e-αnp-n,0e-αmp+m,0+1W¯e-αn-m-e-αne-αmp+n,0p+m,0+Mmn+Nmn,where(28)Mmn=1W¯e-αn-mp-n,αp+m,α-p-n,0p+m,0χn≥m+1,Nmn=1W¯e-αn-mp+n,αp-m,α-p+n,0p-m,0χm≥n.The first term gives a rank one contribution to KE and it diverges as E↑E1, with λ↑∞. The other terms when sandwiched between W1/2 go to finite limits in Hilbert Schmidt norm. The rank one piece gives rise to a bound state near E1 in the weak coupling limit (if ∑‍Wne-2αnp+(n,0)2≤0, even if Wn changes sign). If ∑‍Wne-2αnp+(n,0)2≥0, then the eigenvalue gets absorbed at E=E2 as λ↓0.


## 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.



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