Exactly solvable piecewise analytic double well potential VD(x) = min[(x + d)2, (x − d)2] and its dual single well potential VS(x) = max[(x + d)2, (x − d)2]

Abstract

By putting two harmonic oscillator potentials x2 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 F11 of d and E, which are common to VD and VS but in two different branches. The eigenfunctions are the piecewise square integrable combinations of F11, 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.