ore
10:00
presso Aula Arzelà
Scattering resonances replace bound states/eigenvalues for spectral
problems in which escape (scattering) to infinity is possible. These
states have rates of oscillation and decay and that information is
elegantly encoded in considering the corresponding ``eigenvalues" as
poles of the meromorphic continuation of Green functions. The most
famous ``pure maths" example is given by zeros of the Riemann zeta
function which can be interpreted as resonances for scattering on the
modular surface. In ``applied maths" they appear anywhere from
gravitational waves to MEMS (Micro-Electro-Mechanical Systems).
The mini course will provide a gentle introduction in the setting of
potential scattering in dimension three. Only basic functional
analysis will be a prerequisite.
1. One dimensional scattering: intuition behind outgoing and incoming
waves and the definition of scattering resonances.
2. Analytic Fredholm theory and, as application, meromorphic
continuation of Green's function for potentials scattering in
dimension three.
3. Resonance free regions and expansion of waves in terms of resonances.
4. Counting resonances: upper bounds and existence (and some open
problems). Complex valued potentials with no resonances.
Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math
Sci '17) will provide a reference with a more detailed presentation in
the forthcoming book
http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).