Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let ( M , g ) be an n -dimensional, compact Riemannian manifold and be a semiclassical Schrödinger operator with . Let and be a family of L 2 -normalized eigenfunctions of with . We consider magnetic deformations of of the form , where . Here, u is a k -dimensional parameter running over (the ball o...

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Bibliographic Details
Published in:Annales Henri Poincaré 2013-04, Vol.14 (3), p.611-637
Main Authors: Eswarathasan, Suresh, Toth, John A.
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Dynamical Systems and Ergodic Theory
Elementary Particles
Mathematical and Computational Physics
Mathematical Methods in Physics
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
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Summary:Let ( M , g ) be an n -dimensional, compact Riemannian manifold and be a semiclassical Schrödinger operator with . Let and be a family of L 2 -normalized eigenfunctions of with . We consider magnetic deformations of of the form , where . Here, u is a k -dimensional parameter running over (the ball of radius ), and the family of the magnetic potentials satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of , given by , where for ; the latter functions are themselves eigenfunctions of the -elliptic operators with eigenvalue and . Our main result, Theorem1.2, states that for small, there are constants with j  = 1,2 such that , uniformly for and . We also give an application to eigenfunction restriction bounds in Theorem 1.3.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-012-0198-4