Trace formulae for differential pencils with spectral parameter dependent boundary conditions

Abstract

For the eigenvalues λn of a differential operator, the series MathClass-op∑nλn, generally speaking, diverges; however, it can be regularized by subtracting from λn the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace of Gelfand–Levitan type. A second‐order differential pencil on a finite interval with spectral parameter dependent boundary conditions is considered. We derive the regularized trace formulae of Gelfand–Levitan type for this operator. Copyright © 2013 John Wiley & Sons, Ltd.

Energy dependent potential problems for the one dimensionalp-Laplacian operator

“…where studied in several papers, see [1,2,3,10,22,28,35,36], among other works. The existence of a discrete set of eigenvalues in the linear case is a difficult problem, and it was solved first by Friedman and Shinbrot [11] using functional analytic techniques for linear compact symmetric operators which can be applied to the inverses of the differential operators.…”

A regularized trace formula and oscillation of eigenfunctions of a Sturm‐Liouville operator with retarded argument at 2 points of discontinuity

“…Regularized trace of the Sturm‐Liouville problem with irregular boundary conditions is investigated in Makin . Trace formulas for differential pencils with spectral parameter dependent boundary conditions are obtained in Yang . The first regularized traces of boundary value problems with unbounded operator coefficient in a finite interval are obtained in Maksudov et al and Adiguzelov and Sezer .…”

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

“…Then this work was continued by many authors ( see [1], [5][6][7][8], [12][13][14][15][16][17][18] and [20],respectively). 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].…”