The results exhibited in this thesis are related to Schrodinger operators in three dimensions and are subdivided in two parts based on two published papers, [15] and [14]. A variant of Turbiner's conjecture is proved in the first paper while a partial classification of quasi-exactly solvable Lie algebras of first order differential operators in dimension three is exhibited in the second paper. This classification is then used to construct new quasi-exactly solvable Schrodinger operators in three dimensions. / Turbiner's conjecture posits that, for a Lie algebraic Schrodinger operator in dimension two, the Schrodinger equation is separable if the underlying metric is locally flat. This conjecture is false in general. However, if the generating Lie algebra is imprimitive and if a certain compactness requirement holds, Rob Milson proved that in two dimensions, the Schrodinger equation separates in a Cartesian or polar coordinate system. In [15], the first paper included in this thesis, a similar theorem is proved in three variables. The imprimitivity and compactness hypotheses are still necessary and another condition, related to the underlying metric, must be imposed. In three dimensions, the separation is only partial and the separation will occur in either a spherical, cylindrical or Cartesian coordinate system. / In the second paper [14], a partial classification of quasi-exactly solvable Lie algebras of first order differential operators is performed in three dimensions. Such a classification was known in one and two dimensions but the three dimensional case was still open before the beginning of this research. These new quasi-exactly solvable Lie algebras are used to construct new quasi-exactly solvable Schrodinger operators with the property that part of their spectrum can be explicitly determined. This classification is based on a classification of Lie algebras of vector fields in three variables due to Lie and Amaldi.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.115900 |
| Date | January 2008 |
| Creators | Fortin Boisvert, Mélisande. |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Doctor of Philosophy (Department of Mathematics and Statistics.) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | alephsysno: 002841815, proquestno: AAINR66690, Theses scanned by UMI/ProQuest. |
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