论文标题
可简化的双汇合HEUN方程和原点的一般对称展开
The reducible double confluent Heun equation and a general symmetric unfolding of the origin
论文作者
论文摘要
可简化的双汇合HEUN方程（DCHE）是唯一的DCHE，其对称性展开会导致Fuchsian方程。与通用HEUN方程相反，展开的Fuchsian方程具有5个单数点：$ x_l = - \ sqrt {\ varepsilon}，x_r = \ sqrt {\ varepsilon}，x__ {ll} = - 1/\ sqrt {\ sqrt { x_ {rr} = 1/\ sqrt {\ varepsilon} $和$ 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谐振奇异性$ x_l $周围的矩阵。我们还表明，可简化的DCHE在整个$ \ mathbb {c}^*$中具有全体形态解决方案，并且仅当方程的参数通过第一类的贝塞尔函数连接，并取决于原始的非零chracteristic指数。
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.
