Computing Energy Eigenvalues of Anharmonic Oscillators using the Double Exponential Sinc collocation Method

TL;DR

This paper presents an efficient and highly accurate numerical method combining Sinc collocation with double exponential transformation to compute energy eigenvalues of quantum anharmonic oscillators, including complex multi-well potentials.

Contribution

The authors develop a novel approach that enhances the accuracy and convergence of eigenvalue computations for anharmonic oscillators using an optimized mesh size and the principle of minimal sensitivity.

Findings

Method achieves high accuracy for complex potentials

Convergence properties are rigorously analyzed

Numerical results demonstrate efficiency and precision

Abstract

A quantum anharmonic oscillator is defined by the Hamiltonian ${\cal H}= -\frac{ {\rm d^{2}}}{{\rm d}x^{2}} + V(x)$, where the potential is given by $V(x) = \sum_{i=1}^{m} c_{i} x^{2i}$ with $c_{m}>0$. Using the Sinc collocation method combined with the double exponential transformation, we develop a method to efficiently compute highly accurate approximations of energy eigenvalues for anharmonic oscillators. Convergence properties of the proposed method are presented. Using the principle of minimal sensitivity, we introduce an alternate expression for the mesh size for the Sinc collocation method which improves considerably the accuracy in computing eigenvalues for potentials with multiple wells. We apply our method to a number of potentials including potentials with multiple wells. The numerical results section clearly illustrates the high efficiency and accuracy of the proposed method. All our codes are written using the programming language Julia and are available upon request.