Mathematics Colloquia and Seminars

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It has long been known that in the semiclassical limit $h\to 0$, the number of quantum bound states below a fixed energy $E$ is given asymptotically by $vol(H\leq E)/h^n$, where $H$ is the classical Hamiltonian of the system and $n$ is the number of degrees of freedom. For resonant states in scattering problems, no such estimate is known. I will discuss some numerical results which suggest that the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D\of{K_E}+1}{2}}$. Here, $K_E$ denotes the subset of the classical energy surface $\set{H=E}$ which stays bounded for all time under the flow of $H$ and $D\of{K_E}$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D\of{K_E}+1}{2}$ represents the effective number of degrees of freedom in chaotic scattering problems. This is joint work with Maciej Zworski. Time permitting, I will describe some recent progress on a related problem in the setting of Schottky groups. This is joint work with L. Guillop\'e and M. Zworski.