HIGHLY ACCURATE CALCULATION OF THE REAL AND COMPLEX EIGENVALUES OF ONE-DIMENSIONAL ANHARMONIC OSCILLATORS

Francisco Marcelo Fernández,Javier Garcia

doi:10.14311/ap.2017.57.0391

Abstract

We draw attention on the fact that the Riccati-Padé method developed some time ago enables the accurate calculation of bound-state eigenvalues as well as of resonances embedded either in the continuum or in the discrete spectrum. We apply the approach to several one-dimensional models that exhibit different kind of spectra. In particular we test a WKB formula for the imaginary part of the resonance in the discrete spectrum of a three-well potential.

Highlights

In a recent paper Gaudreau et al[1] proposed a method for the calculation of the eigenvalues of the Schrödinger equation for one-dimensional anharmonic oscillators
We draw attention on the fact that the Riccati-Padé method developed some time ago enables the accurate calculation of bound-state eigenvalues as well as of resonances embedded either in the continuum or in the discrete spectrum
Fd+D fd+D+1 · · · fd+D−1 where D = 2, 3, . . . is the dimension of the determinant and d is the difference between the degrees of the polynomials in the numerator and denominator of the rational approximation to f (x)[3,4,5,6]. In those earlier papers we have shown that there are sequences of roots E[D,d], D = 2, 3, . . . of the determinant HDd (E) that converge towards the bound states and resonances of the quantum-mechanical problem

Summary

Introduction

In a recent paper Gaudreau et al[1] proposed a method for the calculation of the eigenvalues of the Schrödinger equation for one-dimensional anharmonic oscillators. “As can be seen by the numerous approaches which have been developed to solve this problem, there is a beautiful diversity yet lack of uniformity in its resolution While several of these methods yield excellent results for specific cases, it would be favorable to have one general method that could handle any anharmonic potential while being capable of computing efficiently approximations of eigenvalues to a high pre-determined accuracy.”. The authors put forward an approach that they termed double exponential Sinc collocation method (DESCM) and reported results of remarkable accuracy for a wide variety of problems They stated that “In the present work, we use this method to compute energy eigenvalues of anharmonic oscillators to unprecedented accuracy” which may perhaps be true for some of the models chosen but not for other similar examples.

The Riccati-Padé method

Examples

Conclusions