Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator

Let $\mathcal{H}=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$ be the Hermite operator on ${\mathbb{R}}^2$, and let ${\mathrm{\Pi }}_{\lambda }$ denote the spectral projection corresponding to $\lambda =2N+2$. We prove the sharp log-free endpoint estimate $\mathrm{\mid }\mathrm{\mid }{\mathrm{\Pi }}_{\lambda }\mathrm{\mid }{\mathrm{\mid }}_{L^2({\mathbb{R}}^2)\to L^{10\mathrm{/}3}({\mathbb{R}}^2)}\lesssim {\lambda }^{-1\mathrm{/}10}$. 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.