Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: Generalized coherent states for nonlinear algebras

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this...

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Bibliographic Details
Published in:Annals of physics 2018-01, Vol.388, p.26-53
Main Authors: Rosas-Ortiz, Oscar, Zelaya, Kevin
Format: Article
Language:English
Subjects:
ANNIHILATION OPERATORS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Coherent states
EIGENSTATES
EIGENVECTORS
HAMILTONIANS
Harmonic oscillator
HARMONIC OSCILLATORS
HERMITIAN OPERATORS
INTERACTIONS
MULTI-PHOTON PROCESSES
Non-Hermitian operators
Nonlinear algebras
NONLINEAR PROBLEMS
OSCILLATORS
Real spectrum
SPECTRA
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Summary:A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.
ISSN:0003-4916
1096-035X
DOI:10.1016/j.aop.2017.10.020