On Painleve VI transcendents related to the Dirac operator on the hyperbolic disk

TL;DR

This paper explores the spectral properties of the Dirac operator on the hyperbolic disk with magnetic flux, connecting it to Painleve VI transcendents through tau functions and isomonodromic deformations.

Contribution

It explicitly determines the spectrum, resolvent, and Green functions of the Dirac operator, linking these to Painleve VI solutions via tau functions and deformation theory.

Findings

Derived explicit Green functions for the Dirac operator.

Connected the tau function to Painleve VI transcendents with specific parameters.

Analyzed asymptotic behavior of solutions as parameters approach critical values.

Abstract

Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincare disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painleve VI equation with $\gamma=0$. We analyze long distance behaviour of the tau function, as well as the asymptotics of the corresponding Painleve VI transcendents as $s\to 1$. Considering the limit of flat space, we also obtain a class of solutions of the Painleve V equation with $\beta=0$.