  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.
