Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/0104035 · Journal of Physics A Mathematical and General

Generating converging eigenenergy bounds for the discrete states of the -ix3non-Hermitian potential

Carlos R. Handy

Published May 2, 2001• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/34/19/102

Related papers

Moment-method analysis of the ground state of discretized bosonic systems

Carlos R. Handy, Jian Qun Pei

The eigenvalue moment method for bosonic systems, recently developed by C. R. Handy and D. Bessis [Phys. Rev. Lett 55, 931 (1985)], is extended to discrete quantum mechanics. The relevant formalism is described and applied in the context of the discretized harmonic and sextic anharmonic potentials. Rapidly converging bounds to the associated ground-state energies are obtained for fixed lattice spacing a satisfying a<O(1).

Physical Review A•Oct 1, 1988•2 citations

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

Miloslav Znojil

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.

arXiv:math-ph/0101027•Jan 24, 2001•1 citation

A numerical and analytical investigation into non-Hermitian Hamiltonians

Gert Jermia Cornelus Wessels

In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.

•Mar 1, 2009•4 citations

Solvable Analogue of V(x)=ix3

Miloslav Znojil

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.

arXiv:quant-ph/0103115•Mathematics•Jan 29, 2026•1 citation

Dirac and Klein–Gordon particles in complex Coulombic fields; a similarity transformation.

Omar Mustafa, Typeset Using Revtex

The observation that the existence of the amazing reality and discreteness of the spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian is reemphasized in the context of the non-Hermitian Dirac and Klein-Gordon Hamiltonians. Complex Coulombic potentials are considered. In one of the first explicit studies of the non-Hermitian Schrödinger Hamiltonians, Caliceti et al [1] have considered the imaginary cubic oscillator problem in the context of perturbation theory. They have offered the first rigorous explanation why the spectrum in such a model may be real and discrete. Only many years later, after being quoted as just a mathematical curiosity [2] in the literature, the possible physical relevance of this result reemerged and emphasized [3]. Initiating thereafter an extensive discussion which resulted in the proposal of the so called PT −symmetric quantum mechanics by Bender and Boettcher [4]. The spiritual wisdom of the new formalism lies in the observation that the existence of the real spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian. This observation has offered a sufficiently strong motivation for the continued interest in

•Jan 1, 2003•25 citations

Quantization of the potential amplitude in the one-dimensional Schrödinger equation

B. Ya. Balagurov

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕ n ( x ) and eigenvalues α n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕ n ( x ) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.

Journal of Experimental and Theoretical Physics•Oct 1, 2004•3 citations
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.
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.
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.
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 observation that the existence of the amazing reality and discreteness of the spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian is reemphasized in the context of the non-Hermitian Dirac and Klein-Gordon Hamiltonians. Complex Coulombic potentials are considered. In one of the first explicit studies of the non-Hermitian Schrödinger Hamiltonians, Caliceti et al [1] have considered the imaginary cubic oscillator problem in the context of perturbation theory. They have offered the first rigorous explanation why the spectrum in such a model may be real and discrete. Only many years later, after being quoted as just a mathematical curiosity [2] in the literature, the possible physical relevance of this result reemerged and emphasized [3]. Initiating thereafter an extensive discussion which resulted in the proposal of the so called PT −symmetric quantum mechanics by Bender and Boettcher [4]. The spiritual wisdom of the new formalism lies in the observation that the existence of the real spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian. This observation has offered a sufficiently strong motivation for the continued interest in
A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕ n ( x ) and eigenvalues α n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕ n ( x ) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.