The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point

Abstract

We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points.

A trace formula for the Sturm-Liouville type equation with retarded argument

“…Gelfand and Levitan [1] …rstly obtained a trace formula for a self adjoint Sturm-Liouville di¤erential equation. This investigation was continued in many directions, such as Dirac systems [2][3][4], the case of continuous [5][6][7][8][9][10][11], discontinuous [12,13] or matrix Sturm-Liouville operator [14][15][16] and also Sturm-Liouville problems with retarded argument [17][18][19]. In the survey paper [20], the history and the current state of the theory of the regularized trace of the linear operators were presented.…”

The regularized trace of Sturm–Liouville problem with discontinuities at two points

“…A regularized trace formula for Sturm-Liouville equation with one or two boundary conditions depending on a spectral parameter was investigated in [2,4,6,11,22,24]. The regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points was calculated in [9]. All of these works are on the traces of continuous boundary value problems.…”

The inverse spectral problem of some singular version of one-dimensional Schrödinger operator with explosive factor in finite interval

“…From (2.57) and the representation (2.53) and By using the same proof technique of Theorem 2.1 we get the integral equation 3-The present problem, with yð0Þ ¼ 0; yðpÞ ¼ 0 cannot considered as a spacial case of (4.61) 4-Due to the absence of the numbers Hand h the inverse problem by two specters cannot be studied in the present work 5-Besides the direct spectral investigation of the of the present problem as in [14,16], the author had studied the eigenfunction expansion, equiconvergence of the eigenfunction expansion, and the regularized trace formula, [20,21] …”