# 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.
Language English 055002 17 Inverse Problems 27 5 10.1088/0266-5611/27/5/055002 Published - 29 Mar 2011

### Fingerprint

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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title = "A local inverse spectral theorem for Hamiltonian systems",
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.",
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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

Y1 - 2011/3/29

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

U2 - 10.1088/0266-5611/27/5/055002

DO - 10.1088/0266-5611/27/5/055002

M3 - Article

VL - 27

JO - Inverse Problems

T2 - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

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M1 - 055002

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