Direkte und inverse Spektraltheorie periodischer Systeme

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We plan to develop direct and inverse spectral theory in the following cases: A) The matrix Hill operator P=-d2/dx2 + Q, where Q is a symmetric 1-periodic N x N matrix. In this case a first step is the construction of the direct spectral theory: define and analyze the Lyapunov function and develop the theory of the quasimomentum (and the density of states) as a conformal mapping. A second step is to obtain double-sided estimates of the potential in terms of spectral data and to derive consequences for the inverse problem. B) The periodic difference Schrödinger operator. We plan to obtain double-sided estimates of the potential in terms of spectral data and to solve the problem of characterization: Which sets of real intervals can be the spectrum of periodic difference Schrödinger operator? C) The weighted periodic operator T=-p-2d/dx(p2d/dx) + q has zeroes or even changes sign. These problems are associated with the Camassa-Holm equation. D) The Schrödinger operator on the half-line with a compactly supported potential has infinitely many resonances. The problem of characterization is widely open: Which sets of points are the sets of resonances for such an operator? We plan to solve this problem for certain special situations. E) Spectral asymptotics. We plan to determine semiclassical spectral asymptotics for pseudodifferential (periodic and gauge-periodic) problems for certain special situations.