Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$

TL;DR

This paper introduces two exactly solvable quantum potentials, a double well and its dual single well, with eigenvalues and eigenfunctions explicitly determined, revealing tunneling effects through parameter variation.

Contribution

It provides explicit solutions for a class of piecewise analytic double and single well potentials, including their eigenvalues and eigenfunctions, using confluent hypergeometric functions.

Findings

Eigenvalues are zeros of confluent hypergeometric function combinations.

Eigenfunctions are piecewise combinations of confluent hypergeometric functions.

Tunneling effects are visualized by varying the separation parameter d.

Abstract

By putting two harmonic oscillator potential $x^2$ side by side with a separation $2d$, two exactly solvable piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Due to the mirror symmetry, their eigenvalues $E$ for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function ${}_1F_1$ of $d$ and $E$, which are common to $V_{D}$ and $V_{S}$ but in two different branches. The eigenfunctions are the piecewise square integrable combinations of ${}_1F_1$, the so called $U$ functions. By comparing the eigenvalues and eigenfunctions for various values of the separation $d$, vivid pictures unfold showing the tunneling effects between the two wells.