### Abstract

We consider a dissipative evolution problem f_t = iALf - Gf. Here L, G are self-adjoint operators, G>0 has discrete spectrum, and A is a large parameter. The question we would like to ask is under what assumptions on L the dissipation speed is greatly enhanced as the parameter A becomes large. We provide a sharp characterization of operators L that lead to strong enhancement. The work lies at the interface of spectral theory, dynamical systems and PDE. In particular, the original motivation for this study comes from passive scalar equation, where enhancement of diffusion by fluid flow is a classical problem of interest. The key tools invlove estimates related to self adjoint quantum dynamics corresponding to continuous and point spectra. This is a joint work with P. Constantin, L. Ryzhik and A. Zlatos.