Exactly Solvable Sextic Potential Having Symmetric Triple-Well Structure

TL;DR

This paper introduces a new exactly solvable sextic potential with a symmetric triple-well structure, derived using a novel superpotential approach, revealing unique spectral properties and localization effects relevant for quantum tunneling and instanton studies.

Contribution

It presents a new family of exactly solvable triple-well potentials formed by a linear combination of three functions, expanding the class of solvable Schrödinger equations.

Findings

Energy levels are rational functions of quantum number.

Standard harmonic approximation around the central well is invalid.

Excited state probabilities are localized in outer wells.

Abstract

In this paper, we introduce a family of sextic potentials that are exactly solvable, and for the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method. It was suggested since three decades ago that all "additive" or "translational" shape invariant superpotentials formed by two combination of functions have been found and their list was already exhausted by the well-known exactly solvable potentials that are available in most textbooks and furthermore, there are no others. We have devised a new family of superpotentials formed by a linear combination of three functions (two monomials and one rational) and where the change of parameter function is linear in four parameters. This new family of potentials with superpotential $W(x,A,B,D,G) = Ax^3 + Bx -\frac{Dx}{1+Gx^2}$ will extend the list of exactly solvable Schr\"odinger equations. We have shown that the energy of the bound states is rational in the quantum number. Furthermore, approximating the potential around the central well by a harmonic oscillator, as a usual practice, is not valid. The two outer wells affect noticeably the probability density distribution of the excited states. We have noticed that the populations of the triple-well potentials are localized in the two outer wells. These results have potential applications to explore more physical phenomena such as tunneling effect, and instantons dynamics.