Hypergeometric solutions to Schr\"odinger equations for the quantum Painlev\'e equations

TL;DR

This paper introduces hypergeometric solutions to Schr"odinger equations associated with quantum Painlev"e equations and explores their connection to representation theory and KZ equations.

Contribution

It provides explicit hypergeometric solutions and establishes a representation theoretic correspondence with KZ and confluent KZ equations.

Findings

Hypergeometric solutions for quantum Painlev"e Schr"odinger equations

Representation correspondence with KZ and confluent KZ equations

New links between quantum Painlev"e and integrable systems

Abstract

We consider Schr\"odinger equations for the quantum Painlev\'e equations. We present hypergeometric solutions of the Schr\"odinger equations for the quantum Painlev\'e equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schr\"odinger equations for the quantum Painlev\'e equations and those of the KZ equation or the confluent KZ equations.