# Spectral Properties of Non-Self-Adjoint Perturbations for a Spectral Problem with Involution

Authors: Asylzat A. Kopzhassarova; Alexey L. Lukashov; Abdizhakhan M. Sarsenbi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Full description of Riesz basis property for eigenfunctions of boundary value problems for first order differential equations with involutions is given.


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## Body

## 1. Introduction

Differential equations with involutions were considered firstly in [1]. They are a particular case of functional differential equations that appear in several applications (see, for instance, monographs [2, 3] and papers [4–6]). Different spectral problems for equations of this form were considered in [7, 8].In particular, main questions about the following spectral problem:(1.1)u′(x)=λu(-x),-1<x<1,(1.2)u(-1)=γu(1),
were solved in [7]. Namely, (1.1)-(1.2) is a Volterra operator if and only if γ2=-1; furthermore, (1.1)-(1.2) is self-adjoint if and only if γ is a real number. For γ2≠-1, the system of eigenfunctions for (1.1)-(1.2) is a Riesz basis in L2(-1,1). Observe also that for γ2≠-1, (1.1)-(1.2) has no associated functions, that is, all eigenvalues are simple. Note that problem (1.1)-(1.2) is an example of a generalized spectral problem of the form Au=λSu, with  A=d/dx and (Sf)(x)=f(-x).  In general, they were considered in [9] when A and S are operators in a Banach space. Equiconvergence questions for two different perturbations of (1.1)-(1.2) were deeply studied in [10]. The main goal of the paper is to study questions about Riesz basis property of eigenfunctions for the following non-self-adjoint spectral problem:
(1.3)u′(-x)+αu′(x)=λu(x),-1<x<1,(1.4)u(-1)=γu(1).Also note that problems similar to (1.3)-(1.4) appear when the Fourier method is applied for solving boundary value problems for partial differential equations with involution (see, for example, [11] and the bibliography therein).


## 2. Results

Theorem 2.1.
Ifα2≠1, γ≠α±α2-1, then the eigenfunctions system for (1.3)-(1.4) is a Riesz basis in L2(-1,1).Proof.
Before the proof, we need several facts about (1.3)-(1.4). First of all, it is easy to see that the general solution of (1.3)-(1.4) with α2≠1 is given by the following formula:
(2.1)u(x)=C(1+αcos⁡⁡λx1-α2⁡+1-αsin⁡⁡λx1-α2).
Next, we observe that forα2≠1, γ2≠1 eigenvalues are equal to(2.2)λk=1-α2[kπ+arctan1-γ1+γ1+α1-α],k=0,±1,…
The related eigenfunctions are given by the following formula:
(2.3)uk(x)=1+αcos⁡⁡[kπ+arctan1-γ1+γ1+α1-α]x+1-αsin⁡⁡[kπ+arctan1-γ1+γ1+α1-α]x,k=0,±1,…
Observe also that forγ=1, the eigenvalues are
(2.4)λk=1-α2kπ,k=0,±1,±2,…
The corresponding eigenfunctions are
(2.5)uk(x)=1+αcos⁡⁡(kπx)+1-αsin⁡(kπx),k=0,±1,±2,…;
and for γ=-1, the eigenvalues are
(2.6)λk=1-α2(kπ+π2),k=0,±1,…,
and the corresponding eigenfunctions are
(2.7)uk(x)=1+αcos⁡(kπ+π2)x+1-αsin⁡(kπ+π2)x,k=0,±1,…
We introduce the differential operatorL by
(2.8)Lu=u′(-x)+αu′(x),-1<x<1,
and by the boundary condition (1.4). Suppose that Lu belongs to the domain of L, Lu∈D(L). Then we consider
(2.9)L2u=L(u′(-x)+αu′(x))=-(1-α2)u′′(x).
From the boundary condition (1.4), for Lu we deduce that L2 is the second order differential operator generated by the following relations:
(2.10)L2u=-(1-α2)u′′(x),-1<x<1,(2.11)(α-γ)u′(-1)+(1-αγ)u′(1)=0,u(-1)-γu(1)=0.
Spectral problem (2.10)-(2.11) is a typical spectral problem for an ordinary second order differential operator. These problems are studied very well and they have numerous applications (see, for example, [12–14].) Recall [12, Chapter 2] that the following boundary conditions:
(2.12)a1u′(-1)+b1u′(1)+c1u(-1)+d1u(1)=0,c0u(-1)+d0u(1)=0,
for an ordinary second order differential operator are regular if b1c0+a1d0≠0; and they are strongly regular if additionally θ02≠4θ-1θ1, where
(2.13)θ-1=θ1=b1c0+a1d0,θ0=2(a1c0+b1d0).
Since a1=(α-γ), b1=(1-αγ),  c0=1, d0=-γ, we obtain θ-1=θ1=γ2-2αγ+1,   θ0=2(α-2γ+αγ2). It follows from α2≠1, γ2≠1 and γ≠α±α2-1 that the boundary conditions (2.11) are strongly regular. It is known [12, 13] that the eigenfunctions of an operator with strongly regular boundary conditions constitute a Riesz basis in L2(-1,1). By (2.2) numbers -λk cannot be eigenvalues of L, hence any eigenfunction of L2 which corresponds to λk2 will be an eigenfunction L which corresponds to λk as
(2.14)(L2-λk2E)uk=(L+λkE)(L-λkE)uk=0.
Finally, we deduce the assertion of Theorem 2.1 in the case γ2≠1.
For the caseγ2=1 the explicit representations of eigenfunctions (2.5) and (2.7) give the Riesz basis property for these systems directly.Remark 2.2.
Ifα=0, then (1.3)-(1.4) coincide with the unperturbed problem (1.1)-(1.2) which is a Volterra operator for γ2=-1, that is, γ=α±α2-1. If α≠0 and γ=α±α2-1, then the boundary conditions (2.7)–(2.10) are nonregular and hence the system of eigenfunctions is incomplete [12, 13]. Finally, for  α2=1, (1.3) has only trivial solution.
Now, we consider other types of non-selfadjoint perturbations of (1.1)-(1.2).Theorem 2.3.
Ifγ2≠±1, then the eigenfunctions of the following spectral problem:
(2.15)u′(-x)+αu(-x)=λu(x),-1<x<1,u(-1)=γu(1),
constitute a Riesz basis of L2(-1,1).Proof.
The proof is analogous to the proof of Theorem2.1. It uses the following  spectral problem:
(2.16)-u′′(x)+α2u(x)=λu(x),-1<x<1,(2.17)γu′(-1)-u′(1)+α(γ2-1)u(1)=0,u(-1)-γu(1)=0.
Boundary conditions (2.17) are regular for γ2=-1, and nonregular for γ2=1. Then, basis property for eigenfunctions of an ordinary differential second order operator with constant coefficients gives the result for γ2=1.
Forγ2≠±1, boundary conditions (2.17) are strongly regular and the proof terminates analogously to the proof of Theorem 2.1.Remark 2.4.
The perturbationu′(-x)+αu(x)=λu(x) of (1.1)-(1.2) has the same form after the substitution λ-α=μ. Hence, the result of [7] gives full description of basis properties for the following spectral problem:
(2.18)u′(-x)+αu(x)=λu(x),-1<x<1,u(-1)=λu(1).



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