Inverse Spectral Problems for Spectral Data and Two Spectra of N by N Tridiagonal Almost-Symmetric Matrices

Abstract

One way to study the spectral properties of Sturm-Liouville operators is difference equations. The coefficients of the second order difference equation which is equivalent Sturm-Liouville equation can be written as a tridiagonal matrix. One investigation area for tridiagonal matrix is finding eigenvalues, eigenvectors and normalized numbers. To determine these datas, we use the solutions of the second order difference equation and this investigation is called direct spectral problem. Furthermore, reconstruction of matrix according to some arguments is called inverse spectral problem. There are many methods to solve inverse spectral problems according to selecting the datas which are generalized spectral function, spectral data of the matrix and two spectra of the matrix. In this article, we study discrete form the Sturm-Liouville equation with generalized function potential and we will focus on the inverse spectral problems of second order difference equation for spectral data and two spectra. The examined difference equation is equivalent Sturm-Liouville equation which has a discontinuity in an interior point. First, we have written the investigated Sturm-Liouville equation in difference equation form and then constructed N by N tridiagonal matrix from the coefficients of this difference equation system. The inverse spectral problems for spectral data and two-spectra of N by N tridiagonal matrices which are need not to be symmetric are studied. Here, the matrix comes from the investigated discrete Sturm-Liouville equation is not symmetric, but almost symmetric. Almost symmetric means that the entries above and below the main diagonal are the same except two entries.