A numerical and analytical investigation into non-Hermitian Hamiltonians

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dc.contributor.advisorGeyer, H. B.
dc.contributor.advisorWeideman, J. A. C.
dc.contributor.authorWessels, Gert Jermia Cornelus
dc.contributor.otherUniversity of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
dc.date.accessioned2009-02-24T07:38:24Zen_ZA
dc.date.accessioned2010-06-01T09:01:01Z
dc.date.available2009-02-24T07:38:24Zen_ZA
dc.date.available2010-06-01T09:01:01Z
dc.date.issued2009-03
dc.descriptionThesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009.
dc.description.abstractIn 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.en
dc.identifier.urihttp://hdl.handle.net/10019.1/2894
dc.language.isoen
dc.publisherStellenbosch : University of Stellenbosch
dc.rights.holderUniversity of Stellenbosch
dc.subjectNon-Hermitian Hamiltoniansen
dc.subjectHamiltoniansen
dc.subjectDissertations -- Mathematicsen
dc.subjectTheses -- Mathematicsen
dc.subject.lcshSchrodinger equationen
dc.subject.lcshPerturbation (Mathematics)en
dc.titleA numerical and analytical investigation into non-Hermitian Hamiltoniansen
dc.typeThesis
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