Thursday 17 Nov 2016Scarring of Quasimodes on Hyperbolic Manifolds

Suresh Eswarathasan - Cardiff University

H103 14:30-16:30

There is a classical result in microlocal analysis which states that given an elliptic periodic orbit $\gamma$, we can construct quasimodes of order $\mathcal{O}(h^{\infty})$ which concentrate on $\gamma$ (due to Colin de Verdiere, Ralston and others). In the case of hyperbolic orbits, the well-known Gaussian beam construction used to construct these quasimodes breaks down.

Given a compact hyperbolic manifold $(M,g)$ and a totally geodesic submanifold $N$ (or more generally a measurable, flow-invariant subset $\Gamma$), we construct logarithmic quasimodes (i.e. those of order $\epsilon h / |\log h|$) which are partially localized on $N$ (respectively on $\Gamma$). This generalizes previous results of S. Brooks and E.-Nonnenmacher concerning closed orbits on compact surfaces. This is joint work with Lior Silberman (U. British Columbia, Vancouver, Canada).

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