M. Langer and H. Woracek,
A local inverse spectral theorem for Hamiltonian systems,
Inverse Problems 27 (2011), 055002, 17pp.
Abstract:
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.