EVENT DETAILS AND ABSTRACT

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

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

(a) ∫_H | φ_h|² ≥ e^-c_H/h.

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

(b) # { Z _φh ∩ H } = O_H (h^-1).

This is joint work with Yaiza Canzani.