Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

Details Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

2011-12-29

arxiv.org/abs/1112.6213v1

Mathematics

Spectral Theory

19 pages

Scientific paper

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 $\hbar$. We also give an application to eigenfunction restriction bounds.

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