PDE seminar: Absence of embedded eigenvalues for the Schr\”odinger operator on manifold with ends

Date
2012-7-9 16:30 - 2012-7-9 17:30
Place
Faculty of Science Building #3, Room 309
Speaker/Organizer
Kenichi Ito (University of Tsukuba)
 
We consider a Riemannian manifold with, at least,
one expanding end, and prove the absence of
$L^2$-eigenvalues for the Schr\"odinger operator
above some critical value. The critical value is
computed from the volume growth rate of the end
and the potential behavior at infinity. The end
structure is formulated abstractly in terms of some
convex function, and the examples include asymptotically
Euclidean and hyperbolic ends.
The proof consists of a priori superexponential
decay estimate for eigenfunctions and the absence
of superexponentially decaying eigenfunctions, both
of which employs the Mourre-type commutator
argument.

This talk is based on the recent joint work with
E.Skibsted (Aarhus University).