Dirac oscillator in a symmetric sextic anharmonic double-well potential

Abstract

This paper investigates the relativistic Dirac analogue of a one-dimensional Schrödinger oscillator in a symmetric sextic anharmonic double-well potential, constructed via a non-minimal substitution technique. We show that the large component of the Dirac spinor obeys a Schrödinger-like equation, whose eigenvalues are used to determine the relativistic energy levels. These eigenvalues are computed using high-order Bender-Wu perturbation theory, resulting in an algebraic recursion relation for the expansion coefficients. A proper analytic formula describing the large-order behavior of these coefficients is derived, revealing their asymptotic factorial divergence. Additionally, a conjecture illustrating the dependence of these coefficients on the energy quantum numbers is proposed and confirmed. To overcome the challenge posed by the divergent nature of the perturbative eigenvalue expansion, Borel resummation with conformal mapping and Padé approximation are employed, leading to accurate results of the relativistic energy levels and their non-relativistic counterparts as functions of the coupling constants.