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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