## Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions    [PDF]

Suresh Eswarathasan, John A. Toth
Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on \$k\$ parameters \$u\$, for \$k \geq n\$, and satisfies a generic admissibility condition. Define the deformed Schrodinger eigenfunctions to be the \$u\$-parametrized semiclassical family of functions on M that is equal to the unitary magnetic Schrodinger propagator applied to the Schrodinger eigenfunctions. The main result of this article states that the \$L^2\$ norms in \$u\$ of the deformed Schrodinger eigenfunctions are bounded above and below by constants, uniformly on \$M\$ and in \$\hbar\$. In particular, the result shows that this non-random perturbation "kills" the blow-up of eigenfunctions. We give, as applications, an eigenfunction restriction bound and a quantum ergodicity result.
View original: http://arxiv.org/abs/1112.6213