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.
LanguageEnglish
Article number055002
Number of pages17
JournalInverse Problems
Volume27
Issue number5
DOIs
Publication statusPublished - 29 Mar 2011

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Spectral Theorem
Inverse Theorems
Hamiltonians
Hamiltonian Systems
Reparameterization
Uniqueness Theorem
Coefficient
Q-function
Limit Point
Analytic function
Form

Keywords

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

Cite this

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A local inverse spectral theorem for Hamiltonian systems. / Langer, Matthias; Woracek, Harald.

In: Inverse Problems, Vol. 27, No. 5, 055002, 29.03.2011.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A local inverse spectral theorem for Hamiltonian systems

AU - Langer, Matthias

AU - Woracek, Harald

PY - 2011/3/29

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N2 - 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.

AB - 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.

KW - inverse problems

KW - conservation laws

KW - scattering methods

KW - uniqueness theorems

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