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arXiv:quant-ph/0502163 · Modern Physics Letters A

A PERTURBATIVE TREATMENT OF A GENERALIZED $\mathcal{PT}$-SYMMETRIC QUARTIC ANHARMONIC OSCILLATOR

A. Banerjee

Published Nov 24, 2005• 10 citations
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Abstract

We examine a generalized [Formula: see text]-symmetric quartic anharmonic oscillator model to determine the various physical variables perturbatively in powers of a small quantity ε. We make use of the Bender–Dunne operator basis elements and exploit the properties of the totally symmetric operator Tm,n.
DOI
10.1142/s0217732305018475

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A PERTURBATIVE TREATMENT OF A GENERALIZED PT\mathcal{PT}PT-SYMMETRIC QUARTIC ANHARMONIC OSCILLATOR
Two quantum quartic anharmonic many-body oscillators are introduced. One of them is the celebrated Calogero model (rational AnA_nAn​ model) modified by quartic anharmonic two-body interactions which support the same symmetry as the Calogero model. Another model is the three-body Wolfes model (rational G2G_2G2​ model) with quartic anharmonic interaction added which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by the variational method.
We perform a perturbative calculation of the physical observables, in particular, pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator and the classical Hamiltonian for the -symmetric cubic anharmonic oscillator, . Ignoring terms of order 4 and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass, and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum–classical correspondence for this -symmetric system, compute the associated conserved probability density and comment on the issue of factor ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.
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PT\mathcal{P}\mathcal{T}PT -symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
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