Unique continuation properties for the continuous Anderson operator in dimension 2

TL;DR

This paper establishes unique continuation properties for the continuous Anderson operator in 2D, linking its eigenfunctions' nodal sets to Laplace eigenfunctions and applying results to control theory.

Contribution

It introduces a novel approach using quasi-conformal mappings to analyze eigenfunctions and nodal sets of the Anderson operator in low dimensions.

Findings

Eigenfunctions satisfy unique continuation properties.

Nodal sets are quasi-conformal to Laplace eigenfunction nodal sets.

A Courant nodal theorem is proved for the operator.

Abstract

We consider singular continuous Anderson operators $H=\Delta+\xi$ on closed manifolds of dimension 1 and 2, and prove a unique continuation property for its eigenfunctions using the theory of quasi-conformal mappings. We investigate its nodal set by proving that it is quasi-conformal to the nodal set of a Laplace eigenfunction and prove a Courant nodal theorem. We also present an application to control for singular operator in dimension 1.