Title: Scattering Theory: Some Old and New Problems

Scattering theory is, roughly speaking, perturbation theory of self-adjoint operators on the (absolutely) continuous spectrum. It has its origin in mathematical problems of quantum mechanics and is intimately related to the theory of partial differential equations. Some recently solved problems, such as asymptotic completeness for the Schr\"odinger operator with long-range and multiparticle potentials, as well as open problems, are discussed. We construct also potentials for which asymptotic completeness is violated. This corresponds to a new class of asymptotic solutions of the time-dependent Schr\"odinger equation. Special attention is paid to the properties of the scattering matrix, which is the main observable

1991 Mathematics Subject Classification: of the theory. \EndAbstract Primary 35J10, 47A75; Secondary 81U20

Keywords and Phrases: wave operators, asymptotic completeness, the $N$-particle Schr\"odinger operator, new channels of scattering, the scattering matrix

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