Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

Abstract

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

Eigenfunction expansion of the Sturm-Liouville equation with a non-local boundary condition

“…The eigenfunction expansion of nonlocal boundary value problems can be investigate through the method of Green function. For the solution of problem (1.1) when ρ(x) = 1 under different conditions the spectral expansion formula was investigated with different methods in [7]- [9]. In present work we find the eigenfunction expansion formula and prove its convergence for following version of the Sturm-Liouville equation with a non-local boundary condition (1.1)-(1.2).…”

Inverse Sturm–Liouville problems with polynomials in nonseparated boundary conditions

“…The inverse Sturm-Liouville problem for L in the case of separated boundary conditions without polinomials have been well studied (see [5], [8], [9]). The inverse Sturm-Liouville problem for L with spectral parameter in separated boundary conditions have been studied in [1], [3], [6], [7], [10]. The inverse Sturm-Liouville problem with unknown coefficients in nonseparated boundary conditions was studied by V.A.…”

On the solution of an inverse Sturm–Liouville problem with a delay andeigenparameter-dependent boundary conditions

“…Boundary value problems consisting of (1) with α = 1 without eigenparameters in boundary conditions have been studied since the 1930s (for example, see [2,[9][10][11]17,18], and for more details see also [4,7]). In [8,15], inverse Sturm-Liouville problems with boundary conditions depending on an eigenparameter without delay parameter were investigated. Furthermore, in the special case r 1 = r 2 = 0 , problem (1)-( 3) was studied in [5,16].…”