Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator

Abstract

Let $\mathcal{H}=-\Delta +|x{|}^2$ be the Hermite operator on ${\mathbb{R}}^2$, and let ${\Pi }_{\lambda }$ denote the spectral projection corresponding to $\lambda =2N+2$. We prove the sharp log-free endpoint estimate $||{\Pi }_{\lambda }|{|}_{L^2({\mathbb{R}}^2)\to L^{10/3}({\mathbb{R}}^2)}\lesssim {\lambda }^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $T{T}^{\ast }$ argument across radial scales.