Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

TL;DR

This paper establishes uniform bounds on magnetic deformations of Schrödinger eigenfunctions on compact manifolds, showing that such perturbations prevent eigenfunction blow-up and lead to applications in restriction bounds and quantum ergodicity.

Contribution

It provides the first uniform bounds on the $L^2$ norms of deformed eigenfunctions under magnetic perturbations on manifolds.

Findings

Uniform bounds on $L^2$ norms of deformed eigenfunctions.

Magnetic perturbations prevent eigenfunction blow-up.

Applications include eigenfunction restriction bounds and quantum ergodicity.

Abstract

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.