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Inverse problem for Sturm--Liouville operators with frozen argument on closed sets

Maria Kuznetsova

Abstract

In the paper, we study the problem of recovering the potential from the spectrum of the Dirichlet boundary value problem for a Sturm--Liouville equation with frozen argument on a closed set. We consider the case when the closed set consists of two segments and the frozen argument is at the end of the first segment. A uniqueness theorem and an algorithm solving the inverse problem are obtained along with necessary and sufficient conditions of its solvability. The considered case significantly differs from the one of the classical Sturm--Liouville operator with frozen argument.

Inverse problem for Sturm--Liouville operators with frozen argument on closed sets

Abstract

In the paper, we study the problem of recovering the potential from the spectrum of the Dirichlet boundary value problem for a Sturm--Liouville equation with frozen argument on a closed set. We consider the case when the closed set consists of two segments and the frozen argument is at the end of the first segment. A uniqueness theorem and an algorithm solving the inverse problem are obtained along with necessary and sufficient conditions of its solvability. The considered case significantly differs from the one of the classical Sturm--Liouville operator with frozen argument.

Paper Structure

This paper contains 3 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Inverse problem statement. A uniqueness theorem
  3. Necessary and sufficient conditions

Key Result

Lemma 1

Positive zeros of the function $c_2(z^2)$ can be selected to a sequence $\{ z_n\}_{n \ge 1}$ such that the systems $\{ \sin z_n t\}_{n \ge 1}$ and $\{ 1\} \cup \{ \cos z_n t\}_{n \ge 1}$ are Riesz basises in $L_2(0, l).$

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Theorem 2
  • ...and 10 more