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M. Langer, H. Woracek:
"A local inverse spectral theorem for Hamiltonian systems";
Inverse Problems, 27 (2011), 055002.

We consider (2×2)-Hamiltonian systems of the form y

(x) = zJH(x)y(x),
x ∈ [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 de Branges says that the Hamiltonian H
is (up to reparameterization) uniquely determined by the function qH. In this
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.

http://dx.doi.org/10.1088/0266-5611/27/5/055002