In this study, inverse spectral problems for a energy-dependent Sturm-Liouville equations with ?-interaction on a finite interval are considered. Some useful integral representations for the solutions of the considered equation have been derived and using these, properties of the spectral characteristics of the boundary value problem are investigated. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.

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.

In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrix J, which need not be symmetric, by using the GSF and the structure of the GSF is studied.MSC: Primary 39A12; 34A55; 34L15

Inverse spectral and inverse nodal problems are studied for Sturm-Liouville equations with point δ and δ-interactions. Uniqueness theorems are proved and a constructive procedure for the solutions is provided.

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