Inverse problem for Sturm–Liouville differential operators with finite number of constant delays

Mohammad Shahriari

doi:10.3906/mat-2001-80

Abstract

In this manuscript,we study nonself-adjoint second-order differential operators with finite number of constant delays. We investigate the properties of the spectral characteristics and the inverse problem of recovering operators from their spectra. An inverse spectral problem is studied for recovering differential operator from the potential from spectra of two boundary value problems with one common boundary condition.The uniqueness theorem is proved for this inverse problem.

Highlights

The method of separation of variables for solving partial differential equations with finite number of constant delays naturally led to ordinary differential equation with finite number of constant delays inside of the interval which often appear in mathematics, physics, mechanics, geophysics, electronics, meteorology, etc.The interest in differential equations with a constant delay has started intensively growing in the 20th century stimulated by the appearance of various applications in natural sciences and engineering, including the theory of automatic control, the theory of self-oscillating systems, long-term forecasting in the economy, biophysics, etc
The eigenvalues, eigenfunctions, and some properties of the problem are estimated from the known coefficients
The aim of this paper is to investigate the inverse problem of Sturm–Liouville differential equations with finite numbers of constant delays

Summary

Introduction

The method of separation of variables for solving partial differential equations with finite number of constant delays naturally led to ordinary differential equation with finite number of constant delays inside of the interval which often appear in mathematics, physics, mechanics, geophysics, electronics, meteorology, etc.The interest in differential equations with a constant delay has started intensively growing in the 20th century stimulated by the appearance of various applications in natural sciences and engineering, including the theory of automatic control, the theory of self-oscillating systems, long-term forecasting in the economy, biophysics, etc. The inverse spectral Sturm–Liouville problem can be regarded as three aspects: existence, uniqueness, and reconstruction of the coefficients with specific properties of eigenvalues and eigenfunctions, (see [1, 5, 10, 20, 21, 24, 26] and the references therein). The aim of this paper is to investigate the inverse problem of Sturm–Liouville differential equations with finite numbers of constant delays. Some aspects of inverse problems for differential operators with a constant delay were studied in [2, 6, 14, 15, 18, 19, 27, 30].

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Results

Conclusion