A numerical and analytical investigation into non-Hermitian Hamiltonians
Date
2009-03
Authors
Wessels, Gert Jermia Cornelus
Journal Title
Journal ISSN
Volume Title
Publisher
Stellenbosch : University of Stellenbosch
Abstract
In this thesis we aim to show that the Schr odinger 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 Schr odinger
equation with a special class of non-Hermitian Hamiltonians, namely PT -
symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed
by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on
obtaining the real and discrete energy spectrum.
Various methods for solving this problematic Schr odinger equation will
be considered. After starting with perturbation theory, we will move on to
numerical methods. Three di erent 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-Pad e 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 modi cations to the numerical
methods to handle this.
After solving the widely documented PT -symmetric Hamiltonian H =
p2 (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 in
uence on the results.
Description
Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009.
Keywords
Non-Hermitian Hamiltonians, Hamiltonians, Dissertations -- Mathematics, Theses -- Mathematics