Non-Relativistic Quantum Dynamics (eBook)

The bulk of known results in spectral and scattering theory for Schrodinger operators has been derived by time-independent (also called stationary) methods, which make extensive use of re- solvent estimates and the spectral theorem. In very recent years there has been a partial shift of emphasis from the time-indepen- dent to the time-dependent theory, due to the discovery of new, essentially time-dependent proofs of a fair number of the principal results such as asymptotic completeness, absence of singularly con- tinuous spectrum and properties of scattering cross sections. These new time-dependent arguments are somewhat simpler than the station- ary ones and at the same time considerably closer to physical in- tuition, in that they are based on a rather detailed description of the time evolution of states in configUration space (whence the designation "e;geometric methods"e; used by some authors). It seemed interesting to me to present some of these new meth- ods from a strictly time-dependent point of view, by considering as the basic mathematical object strongly continuous unitary one- parameter groups and avoiding the spectral theorem completely. The present volume may be viewed as an essay in this spirit. It is an extended version of a course taught in 1979 at the University of Geneva to undergraduate students enrolled in mathematical physics.