ODE/IM Correspondence in the Semiclassical Limit: Large Degree Asymptotics of the Spectral Determinants for the Ground State Potential

Abstract We study a Schrödinger-like equation for the anharmonic potential $$x^{2 \alpha }+\ell (\ell +1) x^{-2}-E$$ x 2 α + ℓ ( ℓ + 1 ) x - 2 - E when the anharmonicity $$\alpha $$ α goes to $$+\infty $$ + ∞ . When E and $$\ell $$ ℓ 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 $$\ell +\frac{1}{2}$$ ℓ + 1 2 and its zeros converge to the zeros of that Bessel function. We then study the regime in which E and $$\ell $$ ℓ grow large as well, scaling as $$E\sim \alpha ^2 \varepsilon ^2$$ E ∼ α 2 ε 2 and $$\ell \sim \alpha p$$ ℓ ∼ α p . When $$\varepsilon $$ ε 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 $$\frac{2p}{\pi }\frac{\sqrt{\varepsilon ^2-1}}{\varepsilon }$$ 2 p π ε 2 - 1 ε . When $$\varepsilon $$ ε is close to 1 we show that the spectral determinant converges to a function expressed in terms of the Airy function $$\operatorname {Ai}(-)$$ 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.