Discontinuous Sturm–Liouville Problems with Eigenparameter-Dependent Boundary and Transmissions Conditions

Abstract

The purpose of this paper is to study a Sturm-Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but it also appears in both the boundary and transmission conditions.We suggest a new approach for the definition of a suitable Hilbert space and a symmetric linear operator defined in this space in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and for construction and approximation of a fundamental solution. We apply these results to find asymptotic formulas of eigenvalues and corresponding eigenfunctions. (2000): 34L20. Mathematics Subject Classification

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

“…Therefore for the eigenvalue λ n the equality λ n+3 = τ n is true. Moreover, using again Rouche's theorem in γ ε := {λ : λ − λ 0 n < ε}, for sufficiently small ε we get ε n = o (1). Finally the asymptotic formula (8) is obtained via [15] and the equality Δ 0 (λ 0 n + ε n ) = ε n[ Δ 0 λ 0 n + o(1)] holds as ε n → 0.…”

“…[1], [3], [8], [9], [12] and [19] are examples of works with boundary conditions that depend linearly on eigenvalue parameters. Boundary conditions depending nonlinearly on spectral parameter were also considered in [4][5][6], [18], [21][22][23].…”

“…Consider the system of Dirac differential equations y := By + Ω(x)y = λy, x ∈ (0, d) ∪ (d, π), (1) where…”

A class of $2n$th-order differential operator with eigenparameter-dependent boundary and transmission conditions

“…In recent years, highly important results in this field have been obtained for the case when the eigenparameter appears not only in the differential equation but also in the boundary conditions. The literature on such results is voluminous and we refer to [1,2,4,5,7,8,11,13,14]. While the general theory and methods of such second-order boundary value problems are highly developed, very little is known about the general character of the higher order problems.…”

On calculation of eigenvalues and eigenfunctions of a Sturm-Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition

“…In the articles [7][8][9], the asymptotic formulas for the eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions and with the boundary conditions which include spectral parameter were obtained.…”