Accurate energy spectrum for double-well potential: periodic basis

TL;DR

This paper introduces a variational method using periodic basis functions to accurately compute eigenvalues and eigenfunctions of quartic double-well oscillators, outperforming traditional Dirichlet boundary conditions.

Contribution

The study demonstrates that periodic boundary conditions with trigonometric basis functions improve the accuracy of energy spectrum calculations for double-well potentials.

Findings

Periodic basis functions yield higher accuracy than Dirichlet conditions.

The method captures the inflection point in energy graphs effectively.

Periodic boundary conditions allow better fitting to exact solutions.

Abstract

We present a variational study of employing the trigonometric basis functions satisfying periodic boundary condition for the accurate calculation of eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to usual Dirichlet boundary condition, imposing periodic boundary condition on the basis functions results in the existence of an inflection point with vanishing curvature in the graph of the energy versus the domain of the variable. We show that this boundary condition results in a higher accuracy in comparison to Dirichlet boundary condition. This is due to the fact that the periodic basis functions are not necessarily forced to vanish at the boundaries and can properly fit themselves to the exact solutions.