Fractal upper bounds on the density of semiclassical resonances

Abstract

We consider bounds on the number of semiclassical resonances in neighbourhoods of the size of the semiclassical parameter, $h$, around energy levels at which the flow is hyperbolic. We show that the number of resonances is bounded by $h^{-\nu }$, where $2\nu +1$ is essentially the dimension of the trapped set on the energy surface. We note that in a confined setting, this dimension is equal to $2n-1$, where $n$ is the dimension of the physical space and the bound, $h^{1-n}$, corresponds to the optimal bound on the number of eigenvalues. Although no lower bounds of this type are rigorously known in the setting of semiclassical differential operators, the corresponding bound is optimal for certain models based on open quantum maps (see [26])