标题： ODE/IM 对应在半经典极限中：基态势能谱行列式的高次渐进行为 显示英文标题

标题： ODE/IM correspondence in the semiclassical limit: Large degree asymptotics of the spectral determinants for the ground state potential

摘要： 我们研究当非简谐性$\alpha$趋于$+\infty$时，对于非简谐势$x^{2 \alpha}+\ell(\ell+1) x^{-2}-E$的类似薛定谔方程。当$E$和$\ell$在有界域中变化时，我们证明中心连接问题的谱行列式收敛到以阶数为$\ell+\frac{1}{2}$的贝塞尔函数表示的特殊函数，其零点收敛到该贝塞尔函数的零点。 我们接着研究当$E$和$\ell$也变得很大的情况，它们按$E\sim \alpha^2 \varepsilon^2$和$\ell\sim \alpha p$的比例增长。当$\varepsilon$大于$1$时，我们证明中心连接问题的谱行列式是一个快速振荡函数，其零点趋向于按照连续密度定律$\frac{2p}{\pi}\frac{\sqrt{\varepsilon^2-1}}{\varepsilon}$分布。 当$\varepsilon$接近$1$时，我们证明谱行列式收敛到一个用Airy函数$\operatorname{Ai}(-)$表示的函数，并且其零点收敛到该函数的零点。 这项工作受到量子KdV模型的ODE/IM对应关系的启发，并且在该领域有应用。 显示英文摘要

摘要： We study a Schr\"odinger-like equation for the anharmonic potential $x^{2 \alpha}+\ell(\ell+1) x^{-2}-E$ when the anharmonicity $\alpha$ goes to $+\infty$. When $E$ and $\ell$ vary in bounded domains, we show that the spectral determinant for the central connection problem converges to a special function written in terms of a Bessel function of order $\ell+\frac{1}{2}$ and its zeros converge to the zeros of that Bessel function. We then study the regime in which $E$ and $\ell$ grow large as well, scaling as $E\sim \alpha^2 \varepsilon^2$ and $\ell\sim \alpha p$. When $\varepsilon$ is greater than $1$ we show that the spectral determinant for the central connection problem is a rapidly oscillating function whose zeros tend to be distributed according to the continuous density law $\frac{2p}{\pi}\frac{\sqrt{\varepsilon^2-1}}{\varepsilon}$. When $\varepsilon$ is close to $1$ we show that the spectral determinant converges to a function expressed in terms of the Airy function $\operatorname{Ai}(-)$ and its zeros converge to the zeros of that function. This work is motivated by and has applications to the ODE/IM correspondence for the quantum KdV model.