The reducible double confluent Heun equation and a general symmetric unfolding of the origin

TL;DR

This paper studies a special form of the double confluent Heun equation that unfolds into a Fuchsian equation with five singular points, analyzing monodromy and solutions related to Bessel functions.

Contribution

It characterizes the monodromy limits and solution conditions for the reducible double confluent Heun equation's symmetric unfolding.

Findings

Monodromy matrix at the origin is a limit of matrices around other singularities.

Holomorphic solutions exist iff parameters relate via Bessel functions.

Unfolding leads to a Fuchsian equation with five singular points.

Abstract

The reducible double confluent Heun equation (DCHE) is the only DCHE whose general symmetric unfolding leads to a Fuchsian equation. Contrary to general Heun equation the unfolded Fuchsian equation has 5 singular points : $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}, x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$ and $x_{\infty}=\infty$. We prove that the monodromy matrix around the regular resonant singularity at the origin is realizable as a limit of the product of the monodromy matrices around resonant singularities $x_L$ and $x_R$ when $\sqrt{\varepsilon} \to 0$ while the Stokes matrix at the irregular singularity at the origin is a limit of the part of the monodromy matrix around the resonant singularity $x_L$. We also show that the reducible DCHE possesses a holomorphic solution in the whole $\mathbb{C}^*$ if and only if the parameters of the equation are connected by a Bessel function of first kind and order depending on the non-zero chracteristic exponent at the origin.