Zero-range potentials for Dirac particles: Bound-state problems

TL;DR

This paper develops a mathematical model for Dirac particles bound by multiple zero-range potentials, deriving conditions for bound states, and analyzing specific cases with one and two potentials.

Contribution

Introduces a new approach to model Dirac particles with zero-range potentials using boundary conditions parametrized by Hermitian matrices, and derives explicit formulas for bound states.

Findings

Bound states determined by zeros of a specific matrix determinant.

Wave functions are orthogonal under a pseudo-product despite being non-square-integrable.

Explicit Green's function representations for the system are obtained.

Abstract

A model in which a Dirac particle in $\mathbb{R}^{3}$ is bound by $N\geqslant1$ spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle's bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a $2\times2$ Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle's bound-state eigenenergies is reduced to the problem of finding real zeroes of a determinant of a certain $2N\times2N$ matrix. As the lower component of the particle's wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle's matrix Green's function. For illustration purposes, two particular systems are studied in detail: 1) a particle bound in a field of a single zero-range potential, 2) a particle bound in a field of two identical zero-range potentials.