A local inverse spectral theorem for Hamiltonian systems

Matthias Langer, Harald Woracek

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We consider (2×2)-Hamiltonian systems of the form $y'(x) = zJH(x)y(x)$, $x \in [s−, s+)$. If a system of this form is in the limit point case, an analytic function is associated with it, namely its Titchmarsh–Weyl coefficient q_H. The (global) uniqueness theorem due to de Branges says that the Hamiltonian H is (up to reparameterization) uniquely determined by the function q_H. In this paper we give a local uniqueness theorem; if the Titchmarsh–Weyl coefficients q_{H_1} and q_{H_2} corresponding to two Hamiltonian systems are exponentially close, then the Hamiltonians H_1 and H_2 coincide (up to reparameterization) up to a certain point of their domain, which depends on the quantitative degree of exponential closeness of the Titchmarsh–Weyl coefficients.
Original languageEnglish
Article number055002
Number of pages17
JournalInverse Problems
Volume27
Issue number5
DOIs
Publication statusPublished - 29 Mar 2011

Keywords

  • inverse problems
  • conservation laws
  • scattering methods
  • uniqueness theorems

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