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

preprint2022arXivOpen access

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, 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.

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

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

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