Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse

Abstract

In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. 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.

Determination of differential pencils with impulse from interior spectral data

“…It is known [2] that the spectrum of the pencils L(p, q; h, H; a) consists of simple, real eigenvalues λ n , n ∈ Z under the additional assumption that…”

“…such that y(x) = 0 and y (0)y(0)y (π)y(π) = 0. The sequence {λ n } ∞ -∞ satisfies the classical asymptotic form [2]…”

“…China. 2 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, P.R. China.…”

“…The well-known work [4] is the first result about discontinuous inverse eigenvalue problems for the Sturm-Liouville problems, i.e., p(x) ≡ 0 in (1.1). Direct and inverse problems for differential pencils with impulse on a finite interval have been investigated in [2].…”

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

“…Differential equations with potentials depending nonlinearly on the spectral parameter appear frequently in various models of quantum and classical mechanics (see Refs. [1][2][3][4][5][6][7][8] and references there in). For instance, the evolution equations that are used to model interactions between colliding relativistic spineless particles can be reduced to the form (1.1), here the parameter 2 can be regarded as the energy of this system [2,5,6].…”