Unique continuation properties for the continuous Anderson operator in dimension 2

We consider singular continuous Anderson operators $H=\mathrm{\Delta }+\xi$ 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.