24 March 2015, King's College, Strand building, room S5.20.

A special talk:

Maciej Zworski (Berkeley)

Resonances as viscosity limits


Many physical systems can be described using evolution of states. We observe correlations: one measures the time evolution of one state against another state. The time representation can be replaced by the frequency representation (by taking a Fourier transform) which produces the power spectrum. The poles of power spectrum appear in different settings and are called scattering poles (obstacle scattering), quantum resonances (quantum scattering theory), quasinormal modes (general relativity), Pollicott--Ruelle resonances (chaos theory)...

In practically all situations these poles can be defined as limits of L2 eigenvalues of operators which regularize the Hamiltonian at infinity. For instance, Pollicott--Ruelle resonances in the theory of dynamical systems are given by viscosity limits: adding a Laplacian to the generator of an Anosov flow gives an operator with a discrete spectrum; letting the coupling constant go to zero turns eigenvalues into the resonances (joint work with S Dyatlov). This principle seems to apply in all other settings where resonances can be defined and I will explain it in the case of black box Euclidean scattering. The method has also been numerically investigated in the chemistry literature as an alternative to complex scaling.