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

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.