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arXiv:math-ph/0101027

Imaginary cubic oscillator and its square-well approximations in x- and p-representation

Miloslav Znojil

Published Jan 24, 2001• 1 citation
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Abstract

Schroedinger equation with imaginary PT-symmetric potential $V^{}(x) = i\,x^3$ 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.

DOI
10.48550/arxiv.math-ph/0101027

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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.
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.
The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.
Recent investigations by Bender and Boettcher and by Mezincescu have argued that the discrete spectrum of the non-Hermitian potential V ( x ) = -i x 3 should be real. We give further evidence for this through a novel formulation which transforms the general one-dimensional Schrodinger equation (with complex potential) into a fourth-order linear differential equation for |Ψ( x )| 2 . This permits the application of the eigenvalue moment method, developed by Handy, Bessis and coworkers, yielding rapidly converging lower and upper bounds to the low-lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ\ellℓ.