A regularized trace formula for ”weighted” Sturm-Liouville equation with point δ- interaction

Abstract

In this study, we obtain a formula for the regularized trace formula for "weighted" Sturm-Liouville equation with point δ -interaction. At the begining, for the correct determination of solutions of analyzed equation at the point of discontinuty, the matching conditions are required. As a result, an equation is derived for the eigenvalues of the differential operator given in this study.

On the Uniqueness of Inverse Problems for ”weighted” Sturm-Liouville Operator With Delta Interaction

“…For convenience, we denote ρ = √ λ = σ + iτ . It is known [9] that the following asymptotic estimates hold uniformly with respect x ∈ (0, π),…”

“…where q(x) is real-value function in W 1 2 (0, π) and α > 0 ; λ is spectral parameter. It is known [9] that the problem has a discrete spectrum consisting of simple real and bounded below eigenvalues.…”

“…Here we will confine ourselves to the problem of recovering L from the given sets of spectral characteristics. This paper can be viewed as a continuation of [9], in which we gave a self-adjoint operator-theoretic formulation in an appropriate produce space L 2 [0, π] ⊕ C for this problem, obtained some properties of the spectrum and proved some uniqueness theorems for the inverse problems of L.…”

The regularized trace formula for ”weighted” Sturm-Liouville equation with δ-interaction

“…The reqularized trace for differential equations are found in [5]- [11]. However, there is a small numbers of words on the regularized trace for Sturm-Liouville operators with singular potentials (see [2], [13], [15]- [18]). The trace identity of a differential operator deeply reveals spectral structure of the differential operator and has important applications in the numerical calculation of eigenvalues, inverse problem, theory of solutions, and theory of integrable system (see [12], [14]).…”