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Abstract
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This is a short review of the main results on the oscillators considered basic models
in quantum mechanics. Since the potential is polynomial, it is possible to extend
the stationary states to entire functions on the complex plane where the
semiclassical theory works better. The control on the energy levels is based
on the isolation of the nodes, the zeros stable at the unperturbed limit. A
new and simple model for the process of racemization of chiral molecules is
added.
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Keywords
quantum oscillators, $PT$ symmetry, isospectral
Hamiltonians, spectral analysis, analyticity of the
eigenvalues
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Physics and Astronomy Classification Scheme 2010
Primary: 03.65.-w
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Milestones
Received: 25 March 2016
Revised: 29 September 2016
Accepted: 31 October 2016
Published: 4 February 2017
Communicated by Francesco dell'Isola
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