Resonances for asymptotically hyperbolic manifolds: Vasy's method revisited

We revisit Vasy's method for showing meromorphy of the resolvent for (even) asymptotically hyperbolic manifolds. It provides an effective definition of resonances in that setting by identifying them with poles of inverses of a family of Fredholm differential operators. In the Euclidean case the method of complex scaling made this available since the 70's but in the hyperbolic case an effective definition was not known until recently. Here we present a simplified version which relies only on standard pseudodifferential techniques and estimates for hyperbolic operators. As a byproduct we obtain more natural invertibility properties of the Fredholm family.Comment: 22 pages, 1 figur

Scattering resonances as viscosity limits

Using the method of complex scaling we show that scattering resonances of $- \Delta + V$, $V \in L^\infty_{\rm{c}} ( \mathbb R^n )$, are limits of eigenvalues of $- \Delta + V - i \epsilon x^2$ as $\epsilon \to 0+$. That justifies a method proposed in computational chemistry and reflects a general principle for resonances in other settings.Comment: 19 pages, 3 figure

Quantum resonances and partial differential equations

Resonances, or scattering poles, are complex numbers which mathematically describe meta-stable states: the real part of a resonance gives the rest energy, and its imaginary part, the rate of decay of a meta-stable state. This description emphasizes the quantum mechanical aspects of this concept but similar models appear in many branches of physics, chemistry and mathematics, from molecular dynamics to automorphic forms. In this article we will will describe the recent progress in the study of resonances based on the theory of partial differential equations

Stochastic stability of Pollicott-Ruelle resonances

Pollicott-Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.Comment: 26 pages, 6 figures. Added several negative examples at the end of the introductio

Fractal uncertainty for transfer operators

We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists $\sigma > 0$ for which the Selberg zeta function for a convex co-compact hyperbolic surface has only finitely many zeros with $\Re s \geq \frac12 - \sigma$. That eliminates advanced microlocal techniques of Dyatlov-Zahl [arXiv:1504.06589] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature.Comment: 25 pages, 5 figures; minor revisions. To appear in IMR

Quantum decay rates in chaotic scattering

In this article we prove that for a large class of operators, including Schroedinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure. Under the same assumptions we also prove a resolvent estimate with a logarithmic loss compared to nontrapping estimates.Comment: 73 pages, 5 figure