Abstract

In this dissertation, we implement canonical quantization within the framework of the so-called Calogero-Painleve 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 Painleve 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.

We then generalize certain integral representations of solutions of quantum Painleve equations to the multi-particle case. These integral representations are in the form of special beta ensembles of eigenvalues and can be constructed for all the Painleve 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). We elucidate the special values of the corresponding parameters appearing in the quantized Calogero-Painleve equations II-VI.