Averaged pointwise bounds for deformations of Schrödinger Eigenfunctions

Eswarathasan, Suresh and Toth, John A. 2013. Averaged pointwise bounds for deformations of Schrödinger Eigenfunctions. Annales Henri Poincaré 14 (3) , pp. 611-637. 10.1007/s00023-012-0198-4

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Abstract

Let (M,g) be an n-dimensional, compact Riemannian manifold and P0(ℏ)=−ℏ2Δg+V(x) be a semiclassical Schrödinger operator with ℏ∈(0,ℏ0] . Let E(ℏ)∈[E−o(1),E+o(1)] and (ϕℏ)ℏ∈(0,ℏ0] be a family of L2-normalized eigenfunctions of P0(ℏ) with P0(ℏ)ϕℏ=E(ℏ)ϕℏ . We consider magnetic deformations of P0(ℏ) of the form Pu(ℏ)=−Δωu(ℏ)+V(x) , where Δωu(ℏ)=(ℏd+iωu(x))∗(ℏd+iωu(x)) . Here, u is a k-dimensional parameter running over Bk(ϵ) (the ball of radius ϵ), and the family of the magnetic potentials (wu)u∈Bk(ϵ) 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 (ϕℏ)ℏ∈(0,ℏ0] , given by (ϕuℏ)ℏ∈(0,ℏ0] , where ϕ(u)ℏ:=e−it0Pu(ℏ)/ℏϕℏ for |t0|∈(0,ϵ) ; the latter functions are themselves eigenfunctions of the ℏ -elliptic operators Qu(ℏ):=e−it0Pu(ℏ)/ℏP0(ℏ)eit0Pu(ℏ)/ℏ with eigenvalue E(ℏ) and Q0(ℏ)=P0(ℏ). Our main result, Theorem1.2, states that for ϵ>0 small, there are constants Cj=Cj(M,V,ω,ϵ)>0 with j = 1,2 such that C1≤∫Bk(ϵ)|ϕ(u)ℏ(x)|2du≤C2 , uniformly for x∈M and ℏ∈(0,h0] . We also give an application to eigenfunction restriction bounds in Theorem 1.3.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer
ISSN:	1424-0637
Funders:	NSERC
Last Modified:	01 Feb 2020 03:19
URI:	https://orca.cardiff.ac.uk/id/eprint/95087

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