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

2022/10/06 by Tsvetana Stoyanova, Stoyanova, Tsvetana
Chemistry · Mathematics · Physics and Astronomy · #34A25 #34M03 #34M35 #34M40 #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2210.02817

openalex publication_date 2022/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=-√(ε), x_R=√(ε), x_LL=-1/√(ε), x_RR=1/√(ε) and x_∞=∞. 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 √(ε) → 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 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.

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