A new variational perturbation method for double well oscillators

TL;DR

This paper introduces a variational perturbation method for double-well oscillators that achieves high accuracy in energy calculations across various coupling strengths by leveraging eigenvalue spacing and invariant operators.

Contribution

It presents a novel variational perturbation approach based on invariant operators and a natural expansion parameter, improving energy estimates for double-well oscillators.

Findings

Ground and first excited state energies with errors less than 0.01%.

Method applicable over a wide range of coupling strengths.

Provides an iterative formula for higher order corrections.

Abstract

We propose a variational perturbation method based on the observation that eigenvalues of each parity sector of both the anharmonic and double-well oscillators are approximately equi-distanced. The generalized deformed algebra satisfied by the invariant operators of the systems provides well defined Hilbert spaces to both of the oscillators. There appears a natural expansion parameter defined by the ratios of three distance scales of the trial wavefunctions. The energies of the ground state and the first order excited state, in the $0^{th}$ order variational approximation, are obtained with errors $<10^{-2}$% for vast range of the coupling strength for both oscillators. An iterative formula is presented which perturbatively generates higher order corrections from the lower order invariant operators and the first order correction is explicitly given.