PhD Oral Exam - Fatane Mobasheramini, Mathematics
Quantization of Calogero-Painlevé system and Multi-particle quantum Painlevé equations II-VI
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School of Graduate Studies
When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.
Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.
In this dissertation, we implement canonical quantization within the framework of the so-called Calogero-Painlevé correspondence for isomonodromic systems. The classical systems possess a group of symmetries and in the quantum version, we implement the (quantum) Hamiltonian reduction using the Harish-Chandra homomorphism. This allows reducing the matrix operators to Weyl-invariant operators on the space of eigenvalues. We then consider the scalar quantum Painlevé equations as Hamiltonian systems and generalize them to multi-particle systems; this allows us to formulate the multi--particle quantum time-dependent Hamiltonians for the Schrödinger equation ℏ∂_t Ψ=H_J Ψ ,J=II,…,VI.
We then generalize certain integral representations of solutions of quantum Painlevé equations to the multi-particle case. These integral representations are in the form of special β ensembles of eigenvalues and can be constructed for all the Painlevé equations except the first one. They play the role, in the quantum world, of rational solutions in the classical world.
These special solutions exist only for particular values of the quantum Hamiltonian reduction parameter (or coupling constant) κ. We elucidate the special values of the corresponding parameters appearing in the quantized Calogero-Painlevé equations II-VI.