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Scattering on compact manifolds with infinitely thin horns. (English) Zbl 1061.58025

Summary: The quantum-mechanical scattering on a compact manifold with semi-axes attached to the manifold (“hedgehog-shaped manifold”) is considered. The complete description of the spectral structure of Schrödinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven. It is shown that the positive part of the spectrum of the Schrödinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half-line. Moreover, the positive part of the spectrum of the initial Schrödinger operator is fully determined by the conductance properties of an “electronic device” consisting of the initial manifold and two “wires” attached to it.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
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