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Superintégrabilité avec intégrales d'ordre trois, algèbres polynomiales et mécanique quantique supersymétrique
| Content Provider | Semantic Scholar |
|---|---|
| Author | Marquette, Ian |
| Copyright Year | 2009 |
| Abstract | The purpose of my thesis is to pursue a systematic study of classical and quantum superintegrable potentials with a third order integral of motion. We present results from three articles. In the first article, we consider general Hamiltonian that possess a second and third order integral and construct the cubic Poisson algebra generated by integrals of motion. We present the Casimir operator of this algebra. We apply these results to potentials separable in Cartesian coordinates. We present also in this article the trajectories and show that all bounded trajectories are periodic. In the second article, we consider a quantum superintegrable Hamiltonian system in a two-dimensional space with a scalar potential and one quadratic and one cubic integral and construct the most general cubic algebra generated by the integrals and obtain the Casimir operator. We constructed Fock type representations by means of parafermionic algebras. These results give a method to find the degenerate energy spectrum of quantum superintegrable system. We apply this method to many systems. We also present a study of these quantum potentials from the point of view of supersymmetric quantum mechanics and give the spectrum and the eigenfunctions. Finally, in the third article we consider a superintegrable Hamiltonian involving the fourth Painlevé transcendent. We present a study of this system using the cubic algebra generated by its integrals of order 2 and three. We study this system using supersymmetric quantum mechanics with second order supercharge operators and third order shape invariance. We give the energy spectrum and the eigenfunctions. This thesis and the articles it is based upon have advanced our knowledge of superintegrable systems with a third order integral of motion, their relation to supersymmetry and have developed methods to solve the corresponding Hamilton-Jacobi and Schrödinger equations. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://people.smp.uq.edu.au/IanMarquette/index_files/Theseversionfinale.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
