Pseudo-Hamiltonians

Abstract

After establishing the usefulness and plausibility of the semigroup approximation in the preceding chapters, we study here the semigroup time evolution from the point of view of its infinitesimal generator, which we call pseudo-Hamiltonian. First we discuss the relations of such an operator to the total Hamiltonian of the system under consideration. Section 2 is devoted to the mathematical characterization of pseudo-Hamiltonians, which appear to belong to the class of maximal dissipative operators; we deduce here various criteria under which a given operator is (essentially) maximal dissipative. In Section 3, an important particular class of pseudo-Hamiltonians is treated, namely the Schrödinger-type operators with complex absorptive potentials. Next we study one of the situations where the pseudo-Hamiltonian description often appears in practice: the optical approximation in the many-channel scattering processes. We derive some conditions under which this approximation may be expected to be good. Motivated by this problem, we present in the last section some fundamental notions of the non-unitary scattering theory.

“The virtue of the model is that it eliminates a large number of degrees of freedom and leaves one with a more analytically tractable problem.’

E. B. Davies