Scale-free quantitative unique continuation principle

This is joint work with Constanza Rojas-Molina.

For many elliptic partial differential equations it is known that a solution cannot vanish of arbitrary order unless it is identically zero. This can be proven using a Carleman estimate. The latter is also useful to prove a quantitative version of the unique continuation principle.

We are interested in a particular quantitative and scale free version of this result: Consider an eigenfunction of a Schroedinger equation on cube of size L with periodic boundary conditions. Assume that the $L^2$-norm of the eigenfunction is one.

The cube of size L can be decomposed into unit cubes. Place in each unit cube arbitrarily a ball with fixed but small radius. We wnat to derive a lower bound on the $L^2$-norm of the eigenfunction when integrated over the union of the small balls, which is independent of the size $L$, and depends in an explicit way on the other parameters of enterig the problem.