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A class of higher order inverse spectral problems

Ai-Wei Guan, Chuan-Fu Yang, Natalia P. Bondarenko

Abstract

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl-Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

A class of higher order inverse spectral problems

Abstract

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl-Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.
Paper Structure (9 sections, 12 theorems, 79 equations)

This paper contains 9 sections, 12 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Weyl-Yurko matrix
  5. Problems $\mathcal{L}$ and $\mathcal{L}^{\star}$
  6. Asymptotics
  7. Auxiliary Lemma
  8. Recovery from the Weyl functions
  9. Proofs of the uniqueness theorems

Key Result

Lemma 2.1

For $1\leq k\leq j\leq n$, the multiset $\Lambda_{jk}$ coincides with the eigenvalues (counting with multiplicities) of the boundary value problem $S_{jk}$ for the differential equation eqv under the conditions Moreover, the multiset $\Lambda_{jk}$ uniquely determines the corresponding characteristic function $\Delta_{jk}(\lambda)$.

Theorems & Definitions (21)

  • Lemma 2.1: YurBon2
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.1
  • Proposition 3.1: Bon1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • ...and 11 more