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A local inverse spectral theorem for Hamiltonian systems

Langer, Matthias and Woracek, Harald (2011) A local inverse spectral theorem for Hamiltonian systems. Inverse Problems, 27 (5). ISSN 0266-5611

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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.