An approximation to the Woods-Saxon potential based on a contact interaction

TL;DR

This paper introduces a simplified model of the Woods-Saxon potential using a contact interaction, enabling exact solutions for bound states and resonances, and applies it to nuclear data for specific isotopes.

Contribution

It presents a novel approximation of the Woods-Saxon potential through a contact interaction, allowing analytical solutions and application to nuclear structure modeling.

Findings

Exact solutions for bound states and resonances are obtained.

The model effectively approximates the Woods-Saxon potential.

Results agree with experimental data for $^{132}$Sn and $^{208}$Pb.

Abstract

We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial $\delta$-$\delta'$ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei $^{132}$Sn and $^{208}$Pb with an extra neutron.