We study the Hankel determinant generated by a singularly perturbed Jacobi weight$w(x,t):={(1-x^2)}^{\alpha }{\mathrm{e}}^{-\frac{t}{x^2}},x\in [-1,1],\alpha >0,t\ge 0.$ If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor ${\mathrm{e}}^{-\frac{t}{x^2}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.

In the finite n dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of $R_n(t)$, where $R_n(t)$ is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. $n\to \infty$ and $t\to 0$ such that $s=2n^{2}t$ is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

我们研究由奇摄动的Jacobi权重生成的Hankel行列式$w（X，Ť）：={（\mathrm{1个}-X^2）}^{\alpha }{\mathrm{Ë}}^{-\frac{Ť}{X^2}}，X\in [-\mathrm{1个}，\mathrm{1个}]，\alpha >0，Ť\ge 0。$ 如果 $Ť=0$，它减少到经典的对称Jacobi权重。对于$Ť>0$， 因素 ${\mathrm{Ë}}^{-\frac{Ť}{X^2}}$在原点感应一个无限强的零。该汉克尔行列式与混沌腔中的维格纳时滞分布有关。

在有限n维情况下，我们获得两个辅助量${\mathrm{[R}}_{ñ}（Ť）$ 和 ${\mathrm{[R}}_{ñ}（Ť）$通过使用阶梯算子方法。我们证明汉克尔行列式在${\mathrm{[R}}_{ñ}（Ť）$，在哪里 ${\mathrm{[R}}_{ñ}（Ť）$与特定的PainlevéV超越者密切相关。此外，我们导出了汉克行列式对数导数的二阶非线性微分方程和二阶差分方程。这个量可以在神保三轮-冈本来表示σ特定的PainlevéV的-功能之后，我们考虑一个合适的双尺度下，即汉克尔行列式的渐近$ñ\to \infty$ 和 $Ť\to 0$ 这样 $s=2{ñ}^2Ť$是固定的。基于先前使用库仑流体方法得到的结果，我们获得了缩放后的汉克行列式的大s和小s渐近行为，包括渐近展开中的常数项。