L p eigenfunction bounds for the Hermite operator

Abstract

We obtain $L^p$ eigenfunction bounds for the harmonic oscillator $H = -\Delta + x^2$$H=-\Delta +x^2$ in $\mathbb{R}^n$${ℝ}^n$ and for other related operators, improving earlier results of Thangavelu and of Karadzhov. We also construct suitable counterexamples that show that our estimates are sharp.