I will outline a construction of an exotic solution of the nonlinear
Schrödinger equation that exhibits a big frequency cascade. Recent advances
related to this construction and some open questions will be surveyed.

The Hamiltonians of two and three particles moving on d-dimensional lattice and interacting via pairwise short-range potentials are studied.
The following new results are established:
(i).The existence of eigenvalues for the two-particle Shr\"odinger operators depending on the quasi-momentum.
(ii). Infiniteness the number of eigenvalues(Efimov's effect) of the three-particle Shr\"odinger operators
for the zero value of quasi-momentum and its finiteness for the non-zero values of the quasi-momentum.
(iii).The corresponding asymptotics for the number of eigenvalues.

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.