We study a Schrödinger-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.
