L p eigenfunction bounds for the Hermite operator

Abstract

We obtain L p eigenfunction bounds for the harmonic oscillator $H = -\Delta + x^2$ H = - Δ + 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.