Microlocal methods in spectral and scattering theory

Title: Generic lower bounds on the resonance counting function for manifolds hyperbolic near infinity
Speaker: Peter Perry
Speaker Info: University of Kentucky
Brief Description:
Special Note:

Resonances are poles of the resolvent for the Laplacian on a non-compact manifold with `simple' geometry at infinity. They are the natural analogue of eigenvalues for the Laplacian on a compact manifold: they are closely related to the classical geodesic flow, and determine temporal asymptotic behavior of solutions of the wave equation. A fundamental object in the theory is the resonance counting function N(r), which counts the number of resonances in a disc of radius r centered at a chosen fixed point. Typically it is bounded above by rn where n is the dimension of X. Lower bounds (which imply the existence of resonances) are harder to come by and often depend on Poisson-type formulae for resonances. Here we study the counting function for resonances on a class of conformally compact manifolds with constant negative curvature ``near infinity.'' In particular, we prove optimal lower bounds on the resonance counting function by constructing explicit examples of manifolds whose resonance counting function has the optimal order of growth. We exploit a trace formula due to David Borthwick, previous work of Christiansen and Christian-Hislop on generic lower bounds for the counting function in Schrödinger and Euclidean scattering, and a key gluing construction of Sjöstrand and Zworski which we generalize using elementary estimates for the wave equation on a compact manifold.
Date: Saturday, October 29, 2011
Time: 5:00pm
Where: Lunt 105
Contact Person: Dean Baskin
Contact email: dbaskin@math.northwestern.edu
Contact Phone:
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