Some recent results on contact or point supported potentials

TL;DR

This paper explores contact potentials combining Dirac delta functions and their derivatives, analyzing their spectral properties in one dimension, on spheres, and their application to nuclear physics models.

Contribution

It introduces and studies the spectral properties of delta-delta' contact potentials in various geometries and their application to nuclear physics approximations.

Findings

Energy band dependence on parameters in periodic potentials

Number of bound states related to parameters and dimension

Resonance behavior in nuclear physics models

Abstract

We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $\delta$-$\delta'$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $\delta$-$\delta'$ interaction at each node. The dependence of energy bands with the parameters (coefficients of the deltas) can be computed numerically. We also study the $\delta$-$\delta'$ interaction supported on spheres of arbitrary dimension. The spherical symmetry of this model allows us to obtain rigorous conclusions concerning the number of bound states in terms of the parameters and the dimension. Finally, a $\delta$-$\delta'$ interaction is used to approximate a potential of wide use in nuclear physics, and estimate the total number of bound states as well as the behaviour of some resonance poles with the lowest energy.