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Atlantis studies in mathematical physics

Lecture 12: The Method of Enss. Propagation Estimates. Mourre Method. Kato Smoothness, Elements of Algebraic Scattering Theory

Gianfausto Dell’Antonio

Published Jan 1, 2016• 0 citations
Publisher

Abstract

In this lecture we give more details of an alternative approach to quantum scattering theory, initiated by V. Enss, This approach is based on a geometric analysis of the behavior for \(t \rightarrow \pm \infty \) of the solutions of Schroedinger’s equation for initial data in the subspace of absolute continuity for the hamiltonian H . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.2991/978-94-6239-115-4_12

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In this lecture we give more details of an alternative approach to quantum scattering theory, initiated by V. Enss, This approach is based on a geometric analysis of the behavior for (t \rightarrow \pm \infty ) of the solutions of Schroedinger’s equation for initial data in the subspace of absolute continuity for the hamiltonian H . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract A new approach to Quantum Scattering using the Theory of Sources and Absorbers is developed in these chapters. Existing theories of quantum scattering usually calculate the scattered wavefunction and the scattering cross-section by obtaining the solution to Schrödinger’s equation and ignoring the possibility of representing the problem in terms of the principle of stationary action. The action integral approach has several advantages, including quadratic convergence to the accurate results in the variational approach; also, the symmetry of the Lagrangian is readily manifest. Furthermore, the “near-field” wavefunctions and evanescent wave contributions are invariably ignored. Computational methods using discretization of space require the equivalent of “radiation boundary conditions” at large distances, and these are not cleanly implementable.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.