  We study the Hankel determinant generated by a singularly perturbed Jacobi
weight $$
w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\alpha>0,\;\;t\geq
0. $$ 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 Painlev\'{e} 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
$\sigma$-function of a particular Painlev\'{e} V. Then we consider the
asymptotics of the Hankel determinant under a suitable double scaling, i.e.
$n\rightarrow\infty$ and $t\rightarrow 0$ 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.
