A local inverse spectral theorem for Hamiltonian systems

We consider 2×2–Hamiltonian systems of the form y′(t) = zJH(t)y(t), t ∈ [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. AMS Classification Numbers: 34B05, 34A55; 46E22, 42A82