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

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.