# Principal Functions of Non-Selfadjoint Difference Operator with Spectral Parameter in Boundary Conditions

Authors: Murat Olgun; Turhan Koprubasi; Yelda Aygar
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the
principal functions corresponding
to the eigenvalues and the spectral singularities of the boundary value problem
(BVP)an−1yn−1+bnyn+anyn+1=λyn, n∈ℕ and (γ0+γ1λ)y1+(β0+β1λ)y0=0, where (an) and (bn) are complex sequences, λ is an eigenparameter, and γi, βi∈ℂ for i=0, 1.


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## Body

## 1. Introduction

Let us consider the (BVP)(1.1)-y′′+q(x)y=λ2y,0≤x<∞,y′(0)-hy(0)=0
in L2(ℝ+), where q is a complex-valued function and λ∈ℂ is a spectral parameter and h∈ℂ. The spectral theory of the above BVP with continuous and point spectrum was investigated by Naĭmark [1]. He showed that the existence of the spectral singularities in the continuous spectrum of the BVP. He noted that the spectral singularities that belong to the continuous spectrum are the poles of the resolvents kernel but they are not the eigenvalues of the BVP. Also he showed that eigenfunctions and the associated functions (principal functions) corresponding to the spectral singularities are not the element of L2(ℝ+). The spectral singularities in the spectral expansion of the BVP in terms of principal functions have been investigated in [2]. The spectral analysis of the quadratic pencil of Schrödinger, Dirac, and Klein-Gordon operators with spectral singularities was studied in [3–8]. The spectral analysis of a non-selfadjoint difference equation with spectral parameter has been studied in [9]. In this paper, it is proved that the BVP(1.2)an-1yn-1+bnyn+anyn+1=λyn,n∈N,(1.3)(γ0+γ1λ)y1+(β0+β1λ)y0=0
has a finite number of eigenvalues and spectral singularities with a finite multiplicities if (1.4)sup⁡n∈N[exp⁡(ɛnδ)(|1-an|+|bn|)]<∞
for some ɛ>0 and 1/2≤δ≤1.LetL denote difference operator of second order generated in ℓ2(ℕ) by(1.5)(ly)n=an-1yn-1+bnyn+anyn+1,n∈N
and with boundary condition (1.6)(γ0+γ1λ)y1+(β0+β1λ)y0=0,γ0β1-γ1β0≠0,γ1≠a0-1β0,
where {an}n∈ℕ, {bn}n∈ℕ are complex sequences and an≠0 for all n∈ℕ∪{0} and γi,βi∈ℂ for i=0,1.In this paper, which is extension of [9], we aim to investigate the properties of the principal functions corresponding to the eigenvalues and spectral singularities of the BVP (1.2)-(1.3).


## 2. Discrete Spectrum of (1.2)-(1.3)

Let(2.1)sup⁡n∈N[exp⁡⁡(εnδ)(|1-an|+|bn|)]<∞
for some ɛ>0 and 1/2≤δ≤1. The following result is obtained in [10, 11]: under the condition (2.1), equation (1.2) has the solution(2.2)en(z)=αneinz(1+∑m=1∞Anmeimz),n∈N∪{0}
for λ=2cosz, where z∈ℂ¯+:={z:z∈ℂ,Imz≥0} and αn, Anm are expressed in terms of (an) and (bn) as
(2.3)αn=(∏k=n∞ak)-1,An,1=-∑k=n+1∞bk,An,2=-∑k=n+1∞(1-ak2)+∑k=n+1∞bk∑p=k+1∞bp,An,m+2=∑k=n+1∞(1-ak2)Ak+1,m∑k=n+1∞bkAk,m+1+An+1,m.
Moreover, Anm satisfies(2.4)|Anm|≤C∑k=n+⌈m/2⌉∞(|1-ak|+|bk|),
where ⌈m/2⌉ is the integer part of m/2 and C>0 is a constant. So e(z)={en(z)} is continuous in Imz=0 and analytic in ℂ+:={z:z∈ℂ,Imz>0} with respect to z.Let us definef(z) using (2.2) and the boundary condition (1.3) as
(2.5)f(z)=(γ0+2γ1cos⁡z)e1(z)+(β0+2β1cos⁡z)e0(z).
The function f is analytic in ℂ+, continuous in ℂ¯+, and f(z)=f(z+2π).We denote the set of eigenvalues and spectral singularities ofL by σd(L) and σss(L), respectively. From the definition of the eigenvalues and spectral singularities, we have [12] (2.6)σd(L)={λ:λ=2cos⁡z,z∈P0,F(z)=0},σss(L)={λ:λ=2cos⁡z,z∈[-π2,3π2],F(z)=0}∖{0}.
From (2.2) and (2.5), we get(2.7)f(z)=[γ0+γ1(eiz+e-iz)][α1eiz(1+∑m=1∞A1meimz)]+[β0+β1(eiz+e-iz)][α0(1+∑m=1∞A0meimz)]=α0β1e-iz+γ1α1+α0β0+(γ0α1+α0β1)eiz+γ1α1ei2z+∑m=1∞α0β1A0mei(m-1)z+∑m=1∞(γ1α1A1m+α0β0A0m)eimz+∑m=1∞(γ0α1A1m+α0β1A0m)ei(m+1)z+∑m=1∞γ1α1A1mei(m+2)z.
Let(2.8)F(z)=f(z)eiz=α0β1+(γ1α1+α0β0)eiz+(γ0α1+α0β1)e2iz+γ1α1e3iz+∑m=1∞α0β1A0meimz+∑m=1∞(γ1α1A1m+α0β0A0m)ei(m+1)z+∑m=1∞(γ0α1A1m+α0β1A0m)ei(m+2)z+∑m=1∞γ1α1A1mei(m+3)z;
then the function F is analytic in ℂ+, continuous in ℂ¯+, and F(z)=F(z+2π). It follows from (2.6) and (2.8) that(2.9)σd(L)={λ:λ=2cos⁡z,z∈P0,F(z)=0},σss(L)={λ:λ=2cos⁡z,z∈[-π2,3π2],F(z)=0}∖{0}.Definition 2.1.
The multiplicity of a zero ofF in P is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP (1.2) and (1.3).


## 3. Principal Functions

Letλ1,λ2,…,λp and λp+1,λp+2,…,λq denote the zeros of  F in P0:={z:z∈ℂ,z=x+iy,-π/2≤x≤3π/2,y>0} and [-π/2,3π/2] with multiplicities m1,m2,…,mp and mp+1,mp+2,…,mq, respectively.Definition 3.1.
Letλ=λ0 be an eigenvalue of L. If the vectors y(0),y(1),…,y(s); y(k)={yn(k)}n∈ℕk=0,1,…,s satisfy the equations
(3.1)(ly(0))n-λ0yn(0)=0,(ly(k))n-λ0yn(k)-yn(k-1)=0,k=1,2,...,s;n∈N,
then vector y(0) is called the eigenvector corresponding to the eigenvalue λ=λ0 of L. The vectors y(1),…,y(s) are called the associated vectors corresponding to λ=λ0. The eigenvector and the associated vectors corresponding to λ=λ0 are called the principal vectors of the eigenvalue λ=λ0.The principal vectors of the spectral singularities ofL are defined similarly.We define the vectors(3.2)Vn(k)(λj)=1k!{dkdλkEn(λ)}|λ=λj,k=0,1,…,mj-1;j=1,2,…,p,Vn(k)(λj)=1k!{dkdλkEn(λ)}|λ=λj,k=0,1,…,mj-1;j=p+1,p+2,…,q,
where λ=2cosz,z∈P0, and(3.3){En(λ)}:={en(arccos⁡λ2)},n∈N.
Moreover, if y(λ)={yn(λ)}n∈ℕ is a solution of (1.2), then (dk/dλk)y(λ)={(dk/dλk)yn(λ)}n∈ℕ satisfies(3.4)an-1dkdλkyn-1(λ)+bndkdλkyn(λ)+andkdλkyn+1(λ)=λdkdλkyn(λ)+kdk-1dλk-1yn(λ).
From (3.2) and (3.4), we get that(3.5)(lV(0)(λj))n-λjVn(0)(λj)=0,(lV(k)(λj))n-λjVn(k)(λj)-Vn(k-1)(λj)=0,k=1,2,…,mj-1;j=1,2,…,q.
Consequently, the vectors Vn(k)(λj); k=0,1,…,mj-1, j=1,2,…,p and Vn(k)(λj); k=0,1,…,mj-1, j=p+1,p+2,…,q are the principal vectors of eigenvalues and spectral singularities of L, respectively.Theorem 3.2.
(3.6)Vn(k)(λj)∈l2(N);k=0,1,…,mj-1,j=1,2,…,p,Vn(k)(λj)∉l2(N);k=0,1,…,mj-1,j=p+1,…,q.Proof.
UsingEn(λ)=en(arccos(λ/2)), we obtain that
(3.7){dkdλkEn(λ)}|λ=λj=∑ν=0kCν{dνdλνen(z)}z=zj,n∈N,
where λj=2coszj; zj∈P=P0∪[-π/2,3π/2],j=1,2,…,q; Cν is a constant depending on λj.
From (2.2), we find that
(3.8){dνdzνen(z)}z=zj=αneinzj{(in)ν+∑m=1∞[i(n+m)]νAnmeimzj}=αneinzj(in)ν+αneinzj∑m=1∞[i(n+m)]νAnmeimzj.
For the principal vectors Vn(k)(λj)={V(k)(λj)}n∈ℕ, k=0,1,…,mj-1, j=1,2,…,p, corresponding to the eigenvalues λj=2coszj, j=1,2,…,p, of L, we get
(3.9){dkdλkEn(λ)}|λ=λj=∑ν=0kCν{αneinzj(in)ν+αneinzj∑m=1∞[i(n+m)]νAnmeimzj};
then
(3.10)Vn(k)(λj)=1k!{∑ν=0kCν[αneinzj(in)ν+αneinzj∑m=1∞[i(n+m)]νAnmeimzj]}
for k=0,1,…,mj-1, j=1,2,…,p.
SinceImλj>0, j=1,2,…,p from (3.10) we obtain that
(3.11)∑n=1∞|1k!∑ν=0kCναneinzj(in)ν|2≤1(k!)2[∑n=1∞∑ν=0k|Cν||αn|e-nIm⁡zj|nν|]2≤A(k!)2[∑n=1∞e-nIm⁡zj(1+n+n2+⋯+nk)]2≤A(k!)2(k+1)2(∑n=1∞e-nIm⁡zjnk)2<∞,
where A is a constant. Now we define the function
(3.12)gn(z)=1k!∑ν=0kαneinzj∑m=1∞[i(n+m)]νAnmeimzj,j=1,2,…,p.
From (2.4), we obtain that
(3.13)|gn(z)|≤∑ν=0k|αn|e-nIm⁡zj∑m=1∞|n+m|ν|Anm|e-mIm⁡zj≤|αn|e-nIm⁡zj[∑m=1∞|Anm|e-mIm⁡zj+∑m=1∞(n+m)|Anm|e-mIm⁡zj+⋯+∑m=1∞(n+m)k|Anm|e-mIm⁡zj]<Be-nIm⁡zj,
where B=|αn|∑m=1∞∑ν=0k|Anm|e-mImzj(n+m)ν. Therefore, we have
(3.14)∑n=1∞|gn(z)|2≤∑n=1∞B2e-2nIm⁡zj,j=1,2,…,p<∞.
It follows from (3.11) and (3.14) that Vn(k)(λj)∈ℓ2(ℕ), k=0,1,…,mj-1, j=1,2,…,p.
If we consider (3.10) for the principal vectors corresponding to the spectral singularities λj=2coszj, j=p+1,p+2,…,q, of L and consider that Imzj=0 for the spectral singularities, then we have
(3.15)Vn(k)(λj)=1k!{∑ν=0kCναneinzj(in)ν+αneinzj∑ν=0k∑m=1∞[i(n+m)]νAnmeimzj}
for k=0,1,…,mj-1, j=p+1,p+2,…,q.
SinceImλj=0, j=p+1,…,q from (3.15) we find that
(3.16)1k!∑n=1∞|∑ν=0kCναneinzj(in)ν|2=∞.
Now we define tn(z)=∑ν=0k∑m=1∞[i(n+m)]νAnmeimzj, and using (2.4) we get
(3.17)|tn(z)|≤∑ν=0k∑m=1∞|(n+m)ν||Anm|≤∑ν=0k∑m=1∞(n+m)νC∑k=n+⌈m/2⌉∞(|1-ak|+|bk|)≤C∑ν=0k∑m=1∞(n+m)ν∑k=n+⌈m/2⌉∞exp⁡⁡(-εk)exp⁡⁡(εk)(|1-ak|+|bk|)≤C∑ν=0k∑m=1∞(n+m)νexp⁡[-ε4(n+m)]∑k=n+⌈m/2⌉∞exp⁡⁡(εk)(|1-ak|+|bk|)≤C1∑ν=0k∑m=1∞(n+m)νexp⁡[-ε4(n+m)]=C1e(-ε/4)n∑m=1∞∑ν=0k(n+m)νexp⁡(-ε4m)=Ae(-ε/4)n,
where
(3.18)A=C1∑m=1∞∑ν=0k(n+m)νexp⁡(-ɛ4m).
If we use (3.17), we obtain that
(3.19)1k!∑n=1∞|αneinzj∑ν=0k∑m=1∞[i(n+m)]νAnmeimzj|2≤1k!∑n=1∞αn2A2e-ɛn/2<∞.
So Vn(k)∉ℓ2(ℕ), k=0,1,…,mj-1, j=p+1,p+2,…q.Let us introduce Hilbert spaces(3.20)Hk(N)={y={yn}n∈N:∑n∈N(1+|n|)2k|yn|2<∞},H-k(N)={u={un}n∈N:∑n∈N(1+|n|)-2k|un|2<∞},k=0,1,2,…,
with ∥y∥k2=∑n∈ℕ(1+|n|)2k|yn|2, ∥u∥-k2=∑n∈ℕ(1+|n|)-2k|un|2, respectively. It is obvious that H0(ℕ)=ℓ2(ℕ) and(3.21)Hk+1(N)⫋Hk(N)⫋l2(N)⫋H-k(N)⫋H-(k+1)(N),k=1,2,…Theorem 3.3.
Vn(k)(λj)∈H-(k+1)(ℕ),k=0,1,…,mj-1, j=p+1,…,q.Proof.
From (3.15), we have
(3.22)∑n=1∞(1+|n|)-2(k+1)|1k!∑ν=0kCναneinzj(in)ν|2<∞,∑n=1∞(1+|n|)-2(k+1)|1k!∑ν=0kαneinzj∑m=1∞[i(n+m)]νAnmeimzj|2<∞
for k=0,1,…,mj-1, j=p+1,p+2,…,q. Therefore, we obtain that Vn(k)(λj)∈H-(k+1)(ℕ), k=0,1,…,mj-1, j=p+1,p+2,…,q.Let us choosem0=max{mp+1,mp+2,…,mq}. By Theorem 3.2 and (3.21), we get the following.Theorem 3.4.
Vn(k)(λj)∈H-m0(ℕ),k=0,1,…,mj-1,j=p+1,p+2,…,q.Proof.
The proof of theorem is trivial.



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*Source: 608329-2011-06-16.xml*