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

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 : $x_L=-\sqrt{\epsilon },x_R=\sqrt{\epsilon },x_{LL}=-1/\sqrt{\epsilon },x_{RR}=1/\sqrt{\epsilon }$ 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 $\sqrt{\epsilon }\to 0$ 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 ${\mathbb{C}}^{\ast }$ 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.

arXiv comment26 pages, 3 figures