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Journal of Physics A Mathematical and Theoretical

Solvable -symmetric potentials in higher dimensions

GÉZA LÉVAI

Published Mar 23, 2007• 15 citations
Publisher

Abstract

-symmetric, non-relativistic quantum mechanical potentials are discussed in two and three spatial dimensions. Conditions are formulated under which these potentials are -symmetric and can be solved exactly by the separation of the radial and angular variables. It is found that the angular variables play an essential role in introducing non-Hermiticity via the imaginary potential terms. A simple partially exactly solvable potential is used to demonstrate various aspects of symmetry in both two and three dimensions. Possible generalizations of the results are outlined.

DOI
10.1088/1751-8113/40/15/f02

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We present a general procedure by which solvable non-central potentials can be obtained in 2 and 3 dimensions by the separation of the angular and radial variables. The method is applied to generate solvable non-central PT-symmetric potentials in polar coordinates. General considerations are presented concerning the PT transformation properties of the eigenfunctions, their pseudo-norm and the nature of the energy eigenvalues. It is shown that within the present framework the spontaneous breakdown of PT symmetry can be implemented only in two dimensions.
Non-Hermitian, mathcalPT\\mathcal{PT}mathcalPT -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, mathcalPT\\mathcal{PT}mathcalPT potentials for a non-relativistic particle confined in a circular geometry. We show that the mathcalPT\\mathcal{PT}mathcalPT -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that mathcalPT\\mathcal{PT}mathcalPT breaking in two dimensions has a rich phase diagram, with multiple re-entrant mathcalPT\\mathcal{PT}mathcalPT -symmetric phases.
We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.
Exactly solvable PT\mathcal{PT}PT -symmetric models in two dimensions
Non-Hermitian, PT -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, PT potentials for a non-relativistic particle confined in a circular geometry. We show that the PT -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that PT breaking in two dimensions has a rich phase diagram, with multiple re-entrant PT -symmetric phases.