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, 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.

21 pages, 3 figures

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Cited by in corpus (3)

• Harmonic Oscillator with a Step and its Isospectral Properties
• Exactly solvable piecewise analytic double well potential $V_D(x)=min[(x+d{)}^2,(x-d{)}^2]$ and its dual single well potential $V_S(x)=max[(x+d{)}^2,(x-d{)}^2]$
• Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators