Scattering from point interactions

Pavel Exner, Petr Šeba

E. de Prunelé

A solvable model for non relativistic quantum scattering in three-dimensional space of a particle off several interactions centred at n arbitrary points is given. The interaction at one centre can be very different from the interaction at another. Each interaction centred at a given point can be effective on any partial wave with respect to this point. Each centred interaction is a sum of projectors on a finite number of partial waves relative to the centre. Elementary applications are given.

E. B. Davies

J. F. van Diejen, A. Tip

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 ℋ0, but ‘‘live’’ instead in an apparently nonphysical Pontryagin space of the type Πm=ℋ0⊕Cm■Cm. The physical interpretation of the models lies in the association of a unitary S-operator on ℋ0 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.

Yu-Liang Liu

Using eigenfunctional theory, we study two-impurity scattering in one-dimensional interacting fermion systems, and clearly show the evolution of the systems from ultraviolet fixed point to infrared critical fixed point as increasing the impurity scattering potential. For the single-impurity scattering problem, we obtain the same fermion Green's function as that by previous methods. For two-impurity scattering, we demonstrate that the nonlocal interaction of the fermions produced by impurity scattering depends on not only the scattering potential, but also the distance between these two impurities. As two impurities far away from each other, the fermions are completely reflected by the impurities, and in the region between these two impurities there appear stationary waves; while as these two impurities are very close, the fermions with some discrete momentums can easily transmit through them.