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arXiv:quant-ph/0206085

Pseudo-Hermitian version of the charged harmonic oscillator and its "forgotten" exact solutions

Miloslav Znojil

Published Jun 13, 2002• 1 citation
arXivHTMLPDF

Abstract

An unusual type of the exact solvability is reported. It is exemplified by the Coulomb plus harmonic oscillator in D dimensions after a complexification of its Hamiltonian which keeps the energies real. Infinitely many bound states are found in closed form which generalizes the even-parity harmonic-oscillator states at zero charge. Apparently, the model is halfway between exact and quasi-exact.

DOI
10.48550/arxiv.quant-ph/0206085

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Abstract In this work, we describe certain pseudo-Hermitian extensions of the harmonic and isotonic oscillators, both of which are exactly-solvable models in quantum mechanics. By coupling the dynamics of a particle moving in a one-dimensional potential to an imaginary-valued gauge field, it is possible to obtain certain pseudo-Hermitian extensions of the original (Hermitian) problem. In particular, it is pointed out that the Swanson oscillator arises as such an extension of the quantum harmonic oscillator. For the pseudo-Hermitian extensions of the harmonic and isotonic oscillators, we explicitly solve for the wavefunctions in the position representation and also explore their intertwining relations.
An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.