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arXiv:hep-th/0205181 · Journal of Physics A Mathematical and General

Asymptotic solvability of an imaginary cubic oscillator with spikes

Miloslav Znojil, František Gemperle, Omar Mustafa

Published Jun 28, 2002• 29 citations
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Abstract

For complex potentials V ( x ) = −(i x ) 3 − β 2 (i x ) −2 − 2βδ(i x ) 1/2 which are 𝒫𝒯 symmetric, we show that in β ≫ 1 strong coupling regime the low-lying bound states almost coincide with harmonic oscillators whenever the spectrum remains real (this means, at all δ < δ critical (β) ≈ 1).
DOI
10.1088/0305-4470/35/27/317

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Schroedinger equation with imaginary PT-symmetric potential V(x)=i x3V^{}(x) = i\,x^3V(x)=ix3 is studied using the numerical discretization methods in both the coordinate and momentum representations. In the former case our results confirm that the model generates an infinite number of bound states with real energies. In the latter case the differential equation is of the third order and a square-well, solvable approximation of kinetic energy is recommended and discussed. One finds that in the strong-coupling limit, the exact PT-symmetric solutions converge to their Hermitian predecessors.
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Motivated by the recent doubts concerning the compatibility of the purely imaginary one-dimensional cubic-oscillator model with the standard postulates of quantum mechanics, we propose to replace its potential V(x)=ix3 by an elementary piece-wise constant function. We prove that such a simplified non-Hermitian model with a purely imaginary interaction potential still possesses infinitely many bound states with real energies. These states are shown to coincide, incidentally, with their Hermitian square-well analogues in the strong-coupling limit.