Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution

We study the basis properties of systems of eigenfunctions and associated functions for one kind of generalized spectral problems for a second-order ordinary di ﬀ erential operator.


Introduction
Let us consider the partial differential equation with involution w t t, x αw xx t, x w xx t, −x , −1 < x < 1, t > 0. If the function f x ∈ L 2 −1, 1 , then the question about basis property of eigenfunctions of spectral problem for second-order ordinary differential operator with involution raises. Work of many researchers is devoted to the study of differential equations 1-5 . Various aspects of functionally differential equations with involution are studied in 6, 7 . The spectral problems for the double differentiation operator with involution are studied in 8-11 and the issues Riesz basis property of eigenfunctions in terms of coefficients of boundary conditions were considered.
This kind of spectral problems arises in the theory of solvability of differential equations in partial derivatives with an involution 7, page 265 .
Results presented below are a continuation of studies of one of the authors in 9-11 .

General Boundary Value Problem
In this paper, we study the spectral problem of the form where α, β, γ, η, α i , β i , α ij , β ij are some complex numbers. By direct calculation, one can verify that the square of the operator is in the form

2.3
Since it is assumed that Lu belongs to domain of operator L also, then function Lu satisfies boundary-value conditions 2.2 That is, the operator L 2 is generated by previous differential expression and boundary-value conditions 2.2 and 2.4 . The expression L 2 u is an ordinary differential expression for α 0. Therefore, applying the method in 8-10 we can obtain the following statement the result .
Abstract and Applied Analysis 3 Theorem 2.1. If α 0, then the eigenfunctions of the generalized spectral problem 2.1 and 2.2 form a Riesz basis of the space L 2 −1, 1 in the following cases: The root vectors of operators A and A 2 coincide under some conditions see, for instance, 10 . Therefore, we can consider the square of the operator L which is an ordinary differential operator. It is well known 12-14 that eigenfunctions of ordinary differential operator of even order with strongly regular boundary value conditions form a Riesz basis. As in 10 , from here it is possible to deduce correctness of Theorem 2.1.
This technique is not applicable for a 0 since L 2 u is not an ordinary differential operator. Therefore, we consider this case separately.

General Solution of Special Type Equation
Let the operator L be given by the differential expression with an involution and boundary conditions 2.2 . We consider the spectral problem Lu λu x with periodic, antiperiodic boundary conditions, with the boundary conditions of Dirichlet and Sturm type. In these cases, it is possible to compute all the eigenvalues and eigenfunctions explicitly. The basis of our statements is the following. where λ is the spectral parameter, has the form where A and B are arbitrary complex numbers.
Proof. It is easy to see that functions 3.3 are solutions of 3.2 . Let us prove the absence of other solutions.
Any function u x can be represented as a sum of even and odd functions. Substituting this representation into 3.2 and into −u x αu −x λu −x , we conclude that the functions u 1 x and 3.4

The Dirichlet Problem
Consider the spectral problem 3.2 a 2 / 1 with boundary conditions Note that the spectral problem 3.2 and 4.1 is self-adjoint for real α. We calculate the eigenvalues and eigenfunctions of the Dirichlet problem 3.2 and 4.1 . Using Theorem 3.1, it is easy to see that the spectral problem 3.2 and 4.1 has two sequences of simple eigenvalues.

4.3
Theorem 4.1. If a 2 / 1, then the system of eigenfunctions of the spectral problem 3.2 and 4.1 , which is given above, forms an orthonormal basis of the space L 2 −1, 1 .
Proof. For real values of α, the spectral problem 3.2 and 4.1 is self-adjoint. Therefore, the system 4.1 , as a system of eigenfunctions self-adjoint operator, is an orthonormal. Analogously, the case α 8k 2 0 4k 0 1 / 4k 0 1 , k 0 ∈ Z, is considered. Also note that every orthonormal basis is automatically a Riesz basis.
The system 4.2 does not depend on α, hence Theorem 4.1 is proved.
Abstract and Applied Analysis 5

Periodic and Antiperiodic Problem
Now consider the spectral problem 3.2 with the periodic boundary conditions It follows immediately from Theorem 3.1 that the eigenfunctions of the spectral problem 3.2 and 5.1 are given by They are simple and correspond to the eigenfunctions u k1 x sin kπx, k 0, 1, 2, . . . , u k2 x cos kπx, k 0, 1, 2, . . . .

5.3
Similarly, the eigenvalues and eigenfunctions of the spectral problem with antiperiodic boundary conditions are calculated.

5.6
Theorem 5.1. If α 2 / 1, then the systems of eigenfunctions of the spectral problem 3.2 with periodic or antiperiodic boundary conditions form orthonormal bases of the space L 2 −1, 1 .
The proof is analogous to the proof of Theorem 4.1. Also note that for periodic conditions the eigenfunctions form the classical orthonormal basis of L 2 −1, 1 .
Analogously, it is possible to check that the eigenfunctions of spectral problems 3.