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A numerical and analytical investigation into non-Hermitian Hamiltonians
Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2009
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| _version_ | 1867614070433120256 |
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| access_status_str | Open Access |
| author | Wessels, Gert Jermia Cornelus |
| author2 | Geyer, H. B. |
| author_browse | Geyer, H. B. Wessels, Gert Jermia Cornelus |
| author_facet | Geyer, H. B. Wessels, Gert Jermia Cornelus |
| author_sort | Wessels, Gert Jermia Cornelus |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/2894 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:46:11.731Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2009 |
| publishDateRange | 2009 |
| publishDateSort | 2009 |
| publisher | Stellenbosch : University of Stellenbosch |
| publisherStr | Stellenbosch : University of Stellenbosch |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/2894 A numerical and analytical investigation into non-Hermitian Hamiltonians Wessels, Gert Jermia Cornelus Geyer, H. B. Weideman, J. A. C. University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Non-Hermitian Hamiltonians Hamiltonians Dissertations -- Mathematics Theses -- Mathematics Schrodinger equation Perturbation (Mathematics) Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. 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. 2009-02-24T07:38:24Z 2010-06-01T09:01:01Z 2009-02-24T07:38:24Z 2010-06-01T09:01:01Z 2009-03 Thesis http://hdl.handle.net/10019.1/2894 en University of Stellenbosch application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Non-Hermitian Hamiltonians Hamiltonians Dissertations -- Mathematics Theses -- Mathematics Schrodinger equation Perturbation (Mathematics) Wessels, Gert Jermia Cornelus A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title | A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title_full | A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title_fullStr | A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title_full_unstemmed | A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title_short | A numerical and analytical investigation into non-Hermitian Hamiltonians |
| title_sort | numerical and analytical investigation into non hermitian hamiltonians |
| topic | Non-Hermitian Hamiltonians Hamiltonians Dissertations -- Mathematics Theses -- Mathematics Schrodinger equation Perturbation (Mathematics) |
| url | http://hdl.handle.net/10019.1/2894 |
| work_keys_str_mv | AT wesselsgertjermiacornelus anumericalandanalyticalinvestigationintononhermitianhamiltonians AT wesselsgertjermiacornelus numericalandanalyticalinvestigationintononhermitianhamiltonians |