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

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 ${}_1{F}_1$ 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 ${}_1{F}_1$, 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.

A few typos corrected

References in corpus (4)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Symmetrized exponential oscillator
• Confining non-analytic exponential potential $V(x)=g^2\exp (2|x|)$ and its exact Bessel-function solvability

Cited by in corpus (2)

• Harmonic Oscillator with a Step and its Isospectral Properties
• Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators