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Microlocal properties of scattering matrices. (English) Zbl 1346.58012

Summary: We consider the scattering theory for a pair of operators \(H_0\) and \(H= H_0+ V\) on \( L^2( M, m)\), where \(M\) is a Riemannian manifold, \(H_0\) is a multiplication operator on \(M\), and \(V\) is a pseudodifferential operator of order \(\mu\), \(\mu>1\). We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödinger operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.

MSC:

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