Displaced harmonic oscillator $V\sim \min \,[(x+d)^2,(x-d)^2]$ as a   benchmark double-well quantum model
  For the displaced harmonic double-well oscillator the existence of exact polynomial bound states at certain displacements $d\,$ is revealed. The $N-$plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, Schr\"{o}dinger equation can still be considered ``non-polynomially exactly solvable'' (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin. 
Miloslav Znojil
math-ph math.MP