SCATTERING FROM GENERALIZED POINT INTERACTIONS USING SELF-ADJOINT EXTENSIONS IN PONTRYAGIN SPACES

Generalized point interactions (GPI) appearing formally in (quantum) physics as pseudopotentials are given an operator theoretic basis. It is pointed out that in contrast to ordinary point interactions these GPI models are not self-adjoint generators in the physical Hilbert space H(o), but "live" instead in an apparently nonphysical Pontryagin space of the type PI-m = H(o) + C(m)theta-C(m) The physical interpretation of the models lies in the association of a unitary S-operator on H(o) within a two-space setting. A general formalism for GPI is presented and used to answer questions, which were raised in the literature, on the underlying mathematical structure of some models involving pseudopotentials in higher dimensions. Quantum mechanical scattering from GPI, with the purpose of replacing a spherically symmetric short-range interaction potential in the low-energy regime, is discussed in detail. The possible power of using GPI to enable explicit calculations in quantum mechanical and electromagnetic multiple scattering from distortions that are small relative to the wavelength is briefly mentioned.