# New and efficient method for solving the eigenvalue problem for the two-center shell model with finite-depth potentials

## TL;DR

This paper introduces a novel computational approach combining matrix diagonalization and potential expansion in a basis set to efficiently solve the eigenvalue problem in the two-center shell model with finite-depth potentials.

## Contribution

The paper presents a new method that integrates basis expansion with matrix diagonalization to improve efficiency in solving two-center eigenvalue problems.

## Key findings

- Demonstrated efficiency with a system of two $^{16}$O nuclei.
- Applied method to a sum of Woods-Saxon potentials.
- Achieved accurate eigenvalues with reduced computational effort.

## Abstract

We propose a new method to solve the eigen-value problem with a two-center single-particle potential. This method combines the usual matrix diagonalization with the method of separable representation of a two-center potential, that is, an expansion of the two-center potential with a finite basis set. To this end, we expand the potential on a harmonic oscillator basis, while single-particle wave functions on a combined basis with a harmonic oscillator and eigen-functions of a one-dimensional two-center potential. In order to demonstrate its efficiency, we apply this method to a system with two $^{16}$O nuclei, in which the potential is given as a sum of two Woods-Saxon potentials.