Abstract
By putting two harmonic oscillator potential 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 ₁F₁ 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 ₁F₁, 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.