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On non-acceptibility of variational based calculation in $PT$ symmetrized complex Harmonic oscillators: Direct study

Biswanath Rath

Published Jun 9, 2015• 0 citations
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Abstract

We notice through a direct calculation that any variational based calculation on $PT$ symmetrized complex Harmonic Oscillator can lead to breakdown of $PT$ symmetry condition on real spectra. Two different types of oscillators have been tested yielding an uniform conclusion.

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On non-acceptibility of variational based calculation in PTPTPT symmetrized complex Harmonic oscillators: Direct study
We notice through a direct calculation that any variational based calculation on PTPTPT symmetrized complex Harmonic Oscillator can lead to breakdown of PTPTPT symmetry condition on real spectra. Two different types of oscillators have been tested yielding an uniform conclusion.
We notice contradictory statements of Z.ahmed on PT\mathcal{PT}PT symmetry potential V=x24+igx∣x∣V=\frac{x^{2}}{4}+igx|x|V=4x2​+igx∣x∣ (g=2) The curve (dash-dot) shows only one zero whose corresponding energy eigenvalue is 1.72.This graphical representation contradicts the discussion (presented below the graph )which claims number of real discrete states between $0\leg\le2.3$ are 3.as g=2g=2g=2 <g=2.3g=2.3g=2.3 ,one should get number of real discrete states 3 instead of 1.However present calculation matrix diagonalisation pethod reflects 3.secondly the statement on complex Harmonic Oscillator VH=x@+ixV_{H}=x^{@}+ixVH​=x@+ix (which entails complete real discrete spectrum) as a singular example is not acceptable as we notice many complex ∣mathcalPT|mathcal{PT}∣mathcalPT symmetric potentials can be constructed ,which would reflect infinitr real discrete energy eigenvalues.
Here we clearly notice that invariance in commutation relation can be associated with complex energy eigenvalues. Hence present work clearly demonstates that as such we, do not have a general eigenvalue condition in Quantum Physics. Here it is shown that existing all the three eigenvalue conditions namely: Hermiticity, PT symmetry and Pseudo-Hermiticity, are not sufficient enough even to explain the spectra of Harmonic Oscillator under non-Hermitian transformation of momentum, co-ordinate or both, on appearance of complex energy eigenvalues. In fact complex energy eigenvalues become an inherent feature of transformed oscillator, when its frequency of oscillation matches with that of variational based calculation.
A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are P T -symmetric and can be either periodic or regular (of the Pöschl–Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.