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arXiv:2511.07179

Planar Dirac equation with radial contact potentials

J. T. Lunardi, S. Salamanca, J. Negro, L. M. Nieto

math-phquant-phSubmitted Nov 10, 2025• Announced Nov 11, 2025
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Abstract

We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.

Comments
40 pages, 11 figures

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We consider the one-dimensional Dirac equation with the most general relativistic contact interaction supported on two points symmetrically located with respect to the origin. In order to determine the shape of the interaction, we use a distributional method, which in the present case is equivalent to the standard method of defining contact interactions by self-adjoint extensions of symmetric operators. The interaction on each of these two points depends on four parameters, each one having a clear physical meaning. We are interested in the scattering and confining properties of this model. We focus our attention on even or odd interactions under parity transformations and investigate the existence of critical and supercritical states, bound states, confinement and scattering resonances for some particular interactions of special interest.
The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.
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We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.
We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring δδδ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called εεε potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.