Semiclassical Cauchy Estimates and Applications

TL;DR

This paper establishes semiclassical Cauchy estimates for solutions to Schrödinger equations on real analytic manifolds and applies these to bound the measure of nodal sets, advancing understanding of quantum wave behavior.

Contribution

It introduces semiclassical Cauchy estimates and applies Donnelly-Fefferman's method to bound nodal set measures for semiclassical Schrödinger solutions.

Findings

Proved semiclassical Cauchy estimates for derivatives.

Derived bounds for Hausdorff measure of nodal sets.

Enhanced understanding of quantum wave nodal structures.

Abstract

In this note, we study solutions to semiclassical Schrodinger equations on a real analytic manifold with a real analytic potential and prove the semiclassical version of Cauchy estimates on derivatives. As an application, we use Donnelly and Fefferman's method to prove the upper and lower bounds for (n-1)-dimensional Hausdorff measure of the nodal sets of the solutions to semiclassical Schrodinger equations.