Special Day on Eigenfunctions of the Laplacian on manifolds

Title: L2-restriction lower bounds for Schrodinger eigenfunctions in classically forbidden regions
Speaker: John Toth
Speaker Info: McGill University
Brief Description:
Special Note:

Let (M,g) be a compact, closed, real-analytic Riemannian manifold and P(h) = -h2 Δg + V be a Schrodinger operator with (g,V) real-analytic. Given a regular energy value E with consider L2-normalized eigenfunctions φh satisfying P(h) φh = ( E + o(1)) φh . We first prove that for any hypersurface H in the forbidden region Ω(E) = { V > E } there exists a constant cH > 0 such that

(a) ∫H | φh|2 ≥ e-cH/h.

We then use the estimate in (a) to prove that when dim M=2 and H is a Cω hypersurface in Ω(E), the nodal set Z φh = { φh = 0 } satifies the bound

(b) # { Z φh ∩ H } = OH (h-1).

This is joint work with Yaiza Canzani.
Date: Saturday, October 25, 2014
Time: 02:30pm
Where: Harris L28
Contact Person: Valentino Tosatti
Contact email: tosatti@math.northwestern.edu
Contact Phone:
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