Exponential Lower Bounds for Quasimodes of Semiclassical Schr\"{o}dinger Operators

TL;DR

This paper establishes sharp exponential lower bounds for quasimodes of semiclassical Schrödinger operators, demonstrating their optimality and providing insights into unique continuation properties in quantum mechanics.

Contribution

It provides the first sharp exponential lower bounds for quasimodes, confirming the optimality of these bounds and the quasimode accuracy in semiclassical analysis.

Findings

Exponential lower bounds for quasimodes are proven and shown to be sharp.

The results confirm the optimality of quasimode accuracy assumptions.

Unique continuation properties are quantitatively characterized in the semiclassical setting.

Abstract

We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes. We also show that these lower bounds are sharp in h, and that, moreover, the hypothesized quasimode accuracy is also sharp.