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International Journal of Modern Physics C

An efficient method to solve the Schrödinger equation with complex PT-symmetric potentials

F.-Z. Rouabhi, Ismahane Ami, R. Mezhoud, R.J. Lombard

Published Oct 7, 2022• 3 citations
Publisher

Abstract

In this work, we present a simple and efficient method to compute numerically the eigenvalues of complex PT-symmetric Hamiltonian. Numerous works have been devoted to such Hamiltonians since the discovery that they admit totally or partially real spectra. To our knowledge, the method we are advocating has not been used in this context. Besides the determination of real eigenvalues, it allows us to observe the symmetry breaking and to calculate the imaginary parts of the energy.

DOI
10.1142/s0129183123500651

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In this work, we present a simple and efficient method to compute numerically the eigenvalues of complex PT-symmetric Hamiltonian. Numerous works have been devoted to such Hamiltonians since the discovery that they admit totally or partially real spectra. To our knowledge, the method we are advocating has not been used in this context. Besides the determination of real eigenvalues, it allows us to observe the symmetry breaking and to calculate the imaginary parts of the energy.
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.
We study complex PT-symmetric potentials, with real eigenvalues, corresponding to a complex coordinate shift (x+ic2) of a real even potential. In this case,the rules to achieve a coherent quantum mechanics are known. They allow the calculation of observables, which are found to be independent on c. This result is illustratedby few analytical or semi analytical examples. On the other hand, trying to test thisproperty numerically faces problems linked to the difficulty of finding the proper solutions of the Schrodinger equation. In particular, the large distance behaviour of the ¨wave functions generates instabilities. As an example, we have studied the (x +ic2)4potential.
The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2V(x)=ix^2V(x)=ix2 for negative xxx and V(x)=−ix2V(x)=-ix^2V(x)=−ix2 for positive xxx is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.