Oscillatory asymptotics for the Airy kernel determinant on two intervals

TL;DR

This paper derives detailed asymptotic formulas for the Airy kernel Fredholm determinant on two intervals, including oscillatory terms expressed via Jacobi theta-functions, advancing understanding of its spectral properties.

Contribution

It provides explicit asymptotic formulas, including oscillatory terms, for the Airy kernel determinant on two intervals, with expressions involving Jacobi theta-functions.

Findings

Explicit asymptotics up to oscillations of order 1

Formulas expressed in terms of Jacobi theta-functions

Enhanced understanding of the spectral behavior of the Airy kernel

Abstract

We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta$-functions.