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Uniform stability for the Hochstadt-Lieberman problem

Natalia P. Bondarenko

Abstract

In this paper, we prove the uniform stability of the Hochstadt-Lieberman problem, which consists in the recovery of the Sturm-Liouville potential on a half-interval from the spectrum and the known potential on the other half-interval. For this purpose, we reduce the half-inverse problem to the complete one for the unknown potential. Our method relies on the uniform stability for the direct and inverse Sturm-Liouville problems, for recovering sine-type functions from their zeros, and the uniform boundedness of Riesz bases of sines and cosines.

Uniform stability for the Hochstadt-Lieberman problem

Abstract

In this paper, we prove the uniform stability of the Hochstadt-Lieberman problem, which consists in the recovery of the Sturm-Liouville potential on a half-interval from the spectrum and the known potential on the other half-interval. For this purpose, we reduce the half-inverse problem to the complete one for the unknown potential. Our method relies on the uniform stability for the direct and inverse Sturm-Liouville problems, for recovering sine-type functions from their zeros, and the uniform boundedness of Riesz bases of sines and cosines.

Paper Structure

This paper contains 4 sections, 8 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs
  4. Discussion

Key Result

Theorem 1.2

For any $(q^{(1)}, h^{(1)}, H^{(1)})$ and $(q^{(2)}, h^{(2)}, H^{(2)})$ in $P_Q$ and the corresponding mixed data $\mathfrak S^{(1)}$ and $\mathfrak S^{(2)}$, the following uniform estimate holds: Thus, the inverse mapping $\mathfrak S \mapsto \bigl( q(x)_{|x \in (0,\pi)}; \, h \bigr)$ is Lipschitz continuous on the set of mixed data corresponding to $(q, h, H) \in P_Q$.

Theorems & Definitions (11)

  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2: Hryn10
  • Corollary 2.3
  • Lemma 2.4
  • Proposition 2.5: Bond24
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 1 more