Pointwise bounds on quasimodes of semiclassical Schrodinger operators in dimension two

Abstract

We prove optimal pointwise bounds on quasimodes of semiclassical Schrodinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical Lp bounds conducted by Koch-Tataru-Zworski. However, we show that their results imply the two dimensional end-point estimate by scaling and localization.