Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

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 $k\ge n$, 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 $\hslash$. 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.

To appear in Annales Henri Poincaré. 23 pages. Background information on semiclassical wavefronts and eigenfunction concentration has been added. Some notational changes made as well. Further changes made were suggested by the referee

References in corpus (4)

• Local and global analysis of eigenfunctions
• Quantum ergodic restriction theorems, II: manifolds without boundary
• The Loschmidt echo in classically chaotic systems: Quantum chaos, irreversibility and decoherence
• L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

Cited by in corpus (2)

• Long-time dynamics of the perturbed Schrödinger equation on negatively curved surfaces
• On the distribution of perturbations of propagated Schrödinger eigenfunctions