Displaced harmonic oscillator $V\sim \min \,[(x+d)^2,(x-d)^2]$ as a benchmark double-well quantum model

Abstract: 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.