Fractal Weyl Laws for Scattering Resonances

The HADES seminar on Tuesday, October 6th will be given by Yonah Borns-Weilvia Zoom from 3:40 to 5 pm.

Speaker: Yonah Borns-Weil

Abstract:It is well-known that on a bounded domain, the number of eigenvalues of the stationary Schrödinger equation in a given interval follow asymptotics known as Weyl laws. In scattering theory however, we work in an unbounded domain, and such operators need no longer have any eigenvalues. Instead, they have complex resonances, which satisfy a general upper bound (but not necessarily a lower bound) due to Sjöstrand that is analogous to the Weyl law. We present this bound, and describe how the proof must change if we instead count the eigenvalues in an h-dependent region. Following this, we present a result due to Sjöstrand and Zworski, which says that the exponent in such a Weyl law can depend on the fractal dimension of a hyperbolic trapped set. At the end, we will discuss what can still be said when the trapped set is not hyperbolic. Along the way, we will attempt to point out many of the standard “tricks” that are commonly used in scattering theory.

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