Pointwise bounds on quasimodes of semiclassical Schrodinger operators in dimension two

TL;DR

This paper establishes optimal pointwise bounds for quasimodes of semiclassical Schrödinger operators in two dimensions, resolving an open problem in the endpoint estimates of semiclassical Lp bounds.

Contribution

It proves the two-dimensional endpoint estimate for quasimodes, building on and extending the results of Koch-Tataru-Zworski through scaling and localization techniques.

Findings

Optimal pointwise bounds are achieved for quasimodes in 2D.

The endpoint estimate is confirmed for semiclassical Schrödinger operators.

The results connect to and extend previous semiclassical Lp bounds studies.

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.