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arXiv:quant-ph/0103115 · Mathematics

Solvable Analogue of V(x)=ix3

Miloslav Znojil

Published Jan 29, 2026• 1 citation
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Abstract

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.

DOI
10.3390/math14030480

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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.
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.
Recently, a class of -invariant quantum mechanical models described by the non-Hermitian Hamiltonian H = p 2 + x 2 (i x ) was studied. It was found that the energy levels for this theory are real for all 0. Here, the limit as is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H = p 2 + x 2 M (i x ) ( M = 1,2,3, ... ) is also studied, and this -symmetric Hamiltonian becomes exactly solvable in the large- limit as well. In effect, what is obtained in each case is a complex analogue of the Hamiltonian for the square-well potential. Expansions about the large- limit are obtained.
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 in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU(1; 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues in contrast with the common belief that hermiticity is a suffcient condition for real spectrum. The imaginary eigenvalues are derived in algebraic method with imaginary-frequency boson operators for the Hamiltonian of inverted potential well. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions seen to be the polynomials of imaginary frequency are generated from the normalized ground-state wave functions, which are spatially non localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which is turnable by the interaction constant. The transformation operator should not be unitary but Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point, where all eigenstates are degenerate with zero eigenvalue and transition from imaginary to real spectra appears. The SU(1; 1) generator S^z\widehat{S}_{z}Sz​ with real eigenvalues determined by the commutation relation of operators, however, is non-Hermitian in the realization of imaginay-frequency boson operators. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of the canonical variables. It becomes by the canonical transformation of variables a real function indicating exactly the one to one quantum-classical correspondence of Hamiltonians.