We prove that the resolvent set of any (possibly singular)
almost periodic Jacobi operator is characterized as the set of all
energies whose associated Jacobi cocycles induce a dominated splitting.
This extends a well-known result by Johnson for Schrödinger operators.

We establish uniform $L^p$ estimates for resolvents of
elliptic self-adjoint differential operators on compact manifolds
without boundary.  We also show that the spectral regions in our
resolvent estimates are optimal in general. Applications to spectral
theory of periodic Schr\"odinger operators and to inverse boundary
problems will be given. This is joint work with Gunther Uhlmann.

We consider discrete quasi-periodic long range operators with Liouvillean frequency. First, based on generalized Gordon type argument, we show that they can be approximated by a sequence of finite range operators which have no point spectrum for any phase. On the other hand, we show that when the potential for the dual model is small, then they can be approximated by a sequence of long range operators which have at least one eigenvalue for each phase in a set of full measure.

Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.