Displaced Harmonic Oscillator V ∼ 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, the Schrödinger 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.