M. Langer, H. Woracek:

"A local inverse spectral theorem for Hamiltonian systems";

in: "ASC Report 43/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

We consider 2×2-Hamiltonian systems of the form y′(t) = zJH(t)y(t),

t 2 [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

qH. The (global) uniqueness theorem due to L. de Branges says that the

Hamiltonian H is (up to reparameterization) uniquely determined by the

function qH. In the present paper we give a local uniqueness theorem:

if the Titchmarsh-Weyl coefficients qH1 and qH2 corresponding to two

Hamiltonian systems are exponentially close, then the Hamiltonians H1

and H2 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.

Hamiltonian system, Titchmarsh-Weyl coefficient, local uniqueness

http://www.asc.tuwien.ac.at/preprint/2009/asc43x2009.pdf

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