Systematic construction of non-autonomous Hamiltonian equations of Painlev\'e-type. III. Quantization

TL;DR

This paper develops a method for deforming quantized Hamiltonians of Painlevé-type systems into self-adjoint operators, ensuring multi-time solutions for Schrödinger equations and establishing quantum canonical maps between different classes.

Contribution

It introduces a systematic approach to deform minimally quantized quasi-Stäckel Hamiltonians into self-adjoint operators satisfying the quantum Frobenius condition, extending classical integrability concepts to quantum systems.

Findings

Constructed self-adjoint quantum operators with multi-time solutions.

Established quantum canonical maps between magnetic and non-magnetic systems.

Extended classical Painlevé-type system methods to quantum Hamiltonians.

Abstract

This is the third article in our series of articles exploring connections between dynamical systems of St\"ackel-type and of Painlev\'e-type. In this article we present a method of deforming of minimally quantized quasi-St\"ackel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schr\"odinger equations posses common, multi-time solutions. As in the classical case, we obtain here both magnetic and non-magnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps between both classes of quantum systems.