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

Abstract

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

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: 𝑢^′(𝑥)()𝑢()(1),=𝜆𝑢−𝑥,−1<𝑥<1,−1=𝛾𝑢 (1.1)(1.2) 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: 𝑢^′(−𝑥)+𝛼𝑢^′(𝑥)(𝑥)𝑢()(1).=𝜆𝑢,−1<𝑥<1,−1=𝛾𝑢 (1.3)(1.4)

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. If 𝛼²≠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: 𝑢(𝑥)=𝐶√1+𝛼cos𝜆𝑥√1−𝛼²+√1−𝛼sin𝜆𝑥√1−𝛼². (2.1)
Next, we observe that for 𝛼²≠1, 𝛾²≠1 eigenvalues are equal to𝜆_𝑘=√1−𝛼²𝑘𝜋+arctan1−𝛾√1+𝛾1+𝛼√1−𝛼,𝑘=0,±1,… (2.2) The related eigenfunctions are given by the following formula: 𝑢_𝑘(𝑥)=√1+𝛼cos𝑘𝜋+arctan1−𝛾√1+𝛾1+𝛼√𝑥+√1−𝛼1−𝛼sin𝑘𝜋+arctan1−𝛾√1+𝛾1+𝛼√1−𝛼𝑥,𝑘=0,±1,… (2.3)
Observe also that for 𝛾=1, the eigenvalues are 𝜆_𝑘=√1−𝛼²𝑘𝜋,𝑘=0,±1,±2,… (2.4) The corresponding eigenfunctions are 𝑢_𝑘(𝑥)=√1+𝛼cos(𝑘𝜋𝑥)+√1−𝛼sin(𝑘𝜋𝑥),𝑘=0,±1,±2,…; (2.5) and for 𝛾=−1, the eigenvalues are 𝜆_𝑘=√1−𝛼²𝜋𝑘𝜋+2,𝑘=0,±1,…, (2.6) and the corresponding eigenfunctions are 𝑢_𝑘(𝑥)=√𝜋1+𝛼cos𝑘𝜋+2𝑥+√1−𝛼sin𝑘𝜋+𝜋2𝑥,𝑘=0,±1,… (2.7)
We introduce the differential operator 𝐿 by 𝐿𝑢=𝑢^′(−𝑥)+𝛼𝑢^′(𝑥),−1<𝑥<1, (2.8) and by the boundary condition (1.4). Suppose that 𝐿𝑢 belongs to the domain of 𝐿, 𝐿𝑢∈𝐷(𝐿). Then we consider 𝐿²𝑢=𝐿𝑢^′(−𝑥)+𝛼𝑢^′(𝑥)=−1−𝛼²𝑢^′′(𝑥). (2.9)
From the boundary condition (1.4), for 𝐿𝑢 we deduce that 𝐿² is the second order differential operator generated by the following relations: 𝐿²𝑢=−1−𝛼²𝑢^′′(𝑥)𝑢,−1<𝑥<1,𝛼−𝛾^′(−1)+1−𝛼𝛾𝑢^′(1)𝑢()(1)=0,−1−𝛾𝑢=0. (2.10)(2.11)
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: 𝑎₁𝑢^′(−1)+𝑏₁𝑢^′(1)+𝑐₁𝑢(−1)+𝑑₁𝑢(1)𝑐=0,₀𝑢(−1)+𝑑₀𝑢(1)=0, (2.12) for an ordinary second order differential operator are regular if 𝑏₁𝑐₀+𝑎₁𝑑₀≠0; and they are strongly regular if additionally 𝜃²₀≠4𝜃₋₁𝜃₁, where 𝜃₋₁=𝜃₁=𝑏₁𝑐₀+𝑎₁𝑑₀,𝜃₀=2𝑎₁𝑐₀+𝑏₁𝑑₀. (2.13) 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 −𝜆_𝑘 cannot be eigenvalues of 𝐿, hence any eigenfunction of 𝐿² which corresponds to 𝜆²_𝑘 will be an eigenfunction 𝐿 which corresponds to 𝜆_𝑘 as 𝐿²−𝜆²_𝑘𝐸𝑢_𝑘=𝐿+𝜆_𝑘𝐸𝐿−𝜆_𝑘𝐸𝑢_𝑘=0. (2.14) Finally, we deduce the assertion of Theorem 2.1 in the case 𝛾²≠1.
For the case 𝛾²=1 the explicit representations of eigenfunctions (2.5) and (2.7) give the Riesz basis property for these systems directly.

Remark 2.2. If 𝛼=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. If 𝛾²≠±1, then the eigenfunctions of the following spectral problem: 𝑢^′()()(𝑥)𝑢()(1),−𝑥+𝛼𝑢−𝑥=𝜆𝑢,−1<𝑥<1,−1=𝛾𝑢 (2.15) constitute a Riesz basis of 𝐿₂(−1,1).

Proof. The proof is analogous to the proof of Theorem 2.1. It uses the following spectral problem: −𝑢^′′(𝑥)+𝛼²𝑢(𝑥)=𝜆𝑢(𝑥),−1<𝑥<1,𝛾𝑢^′(−1)−𝑢^′(1)+𝛼𝛾²𝑢(1)𝑢()(1)−1=0,−1−𝛾𝑢=0. (2.16)(2.17)
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.
For 𝛾²≠±1, boundary conditions (2.17) are strongly regular and the proof terminates analogously to the proof of Theorem 2.1.

Remark 2.4. The perturbation 𝑢^′(−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: 𝑢^′()(𝑥)(𝑥)𝑢()(1).−𝑥+𝛼𝑢=𝜆𝑢,−1<𝑥<1,−1=𝜆𝑢 (2.18)