论文标题
PainlevéV和Hankel的决定因素，jacobi重量很大
Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
论文作者
论文摘要
我们研究由奇异扰动的雅各比（Jacobi）重量$$产生的汉克尔决定因素w（x，t）：=（1-x^2）^α\ mathrm {e}^{ - \ frac {t} {x^{2}}}，\; \; \; \; \; \; \; \; \; x \;减少到经典的对称雅各比的重量。对于$ t> 0 $，因子$ \ mathrm {e}^{ - \ frac {t} {x^{2}}}} $诱导了原始的无限强零。该汉克尔的决定因素与混沌腔中的Wigner时间延迟分布有关。 在有限的$ n $尺寸案例中，我们通过使用梯子操作员方法获得了两个辅助数量$ r_n（t）$和$ r_n（t）$。我们表明，汉克尔决定因素在$ r_n（t）$方面具有不可或缺的表示形式，其中$ r_n（t）$与特定的painlevév verscentent密切相关。此外，我们得出了二阶非线性微分方程，也导出了汉克尔决定簇的对数衍生物的二阶差方程。该数量可以用特定的Pachlevév的Jimbo-Miwa-Okamoto $σ$ function表示。那么，我们考虑了在适当的双重缩放下汉克尔决定簇的渐近学，即$ n \ rightArrow \ rightarrow \ rightarrow \ infty \ infty \ infty \ infty \ infty $ and $ t \ rightarrow 0 $ t \ rightarrow 0 $ 0 $ $ s $ s = 2n^$ = 2n^$ = 2 nes^2 p t yes wirdes。基于使用库仑流体方法的先前结果，我们获得了缩放的汉克尔决定簇的大$ S $和小$ S $渐近行为，包括渐近扩张中的恒定项。
We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;α>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 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. $n\rightarrow\infty$ 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.
