Distributional approach to point interactions in one-dimensional quantum mechanics

Abstract

We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ interaction.

Double General Point Interactions: Symmetry and Tunneling Times

“…According to Calçada et al [39] the distributional Schrödinger equation can be written as (throughout this paper we will adopt the atomic Rydberg units such thath = 1 and 2m = 1, as usual when dealing with point interactions)…”

“…Such interaction terms correspond to non-separated solutions, and will be the only type of interactions considered in this work. The above distributional Schrödinger equation implies the same boundary conditions at the point singularities as those obtained from the SAE method [39,53] (also see [54] for a rigorous study of the boundary conditions for point interactions):…”

“…These controversies, however, were clarified by using mathematically rigorous methods such as selfadjoint extensions (SAE) [34][35][36][37] and Schwartz's distribution theory [38,39] (there are also alternative approaches using non-standard distribution theory, e.g., [40,41]), both of which unequivocally demonstrate the existence of a four-parameter family of point interactions in one dimensional quantum mechanics-these include, but are not restricted to, the δ interaction as well as other interactions commonly associated with the δ ′ in the literature. It should be noticed that the distributional approach [39] makes it unambiguously clear, from parity considerations, that a "genuine" δ ′ interaction does not exist (for a particular regularization of the δ ′ interaction displaying even behavior under parity transformations see [42]). …”

“…Futhermore, in addition to its relevance to the study of resonant tunneling in heterostructures in the zero-range limit [49,50], the presence of two point barriers allows us to investigate the GHE-in contradistinction, the HE cannot be addressed by considering only one point scatterer, since the distance "traveled" in that case vanishes (the support of a group of point interactions is a set of zero measure). Therefore, in this work we investigate the scattering of a non-relativistic particle by two point scatterers given by the most general potential [34][35][36]39]. Given the relevance of parity-symmetric considerations in the controversial discussion of regularized implementations of point interactions, and the related discussion of the δ ′ -interaction, we start by introducing the singular interaction following the distributional approach introduced in Lunardi and Manzoni [39], and analyzing the behavior of such potential under parity transformations, thus obtaining the necessary conditions to have even and odd potentials under such transformations.…”

“…Therefore, in this work we investigate the scattering of a non-relativistic particle by two point scatterers given by the most general potential [34][35][36]39]. Given the relevance of parity-symmetric considerations in the controversial discussion of regularized implementations of point interactions, and the related discussion of the δ ′ -interaction, we start by introducing the singular interaction following the distributional approach introduced in Lunardi and Manzoni [39], and analyzing the behavior of such potential under parity transformations, thus obtaining the necessary conditions to have even and odd potentials under such transformations. We also analyze the limits in which the separation between the barriers shrinks to zero and extend the work of Šeba [35] to include the limit for a renormalized 1 odd-arrangement of two δ ′ -interactions 2 .…”