We study one-dimensional ergodic operator family with sampling function $\{x\}$ and some its generalizations. The general result by Damanik and Killip implies that they cannot have absolute continuous spectra. We show that for almost all frequencies and all coupling constants these operators have pure point spectrum of positive Lebesgue measure, and that singular continuous spectrum is supported on a closed set of measure zero. In addition, there is no singular continuous spectrum for large coupling. The results are joint with Svetlana Jitomirskaya.

 The dry version of the ten Martini Problem is an interesting problem of Almost Mathieu Operator. In  this talk,

we will discuss  the reducibility    of a  Schrodinger cocycle  by the methods of Localization in the dual model.  As an application, we

will   show  AMO has open gaps with small coupling. 

Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains").

 It recently transpired that the difference between this upper bound and the actual value can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: (1) stability of the nodal partitions of a domain in R^d with respect to a perturbation of the partition boundaries and (2) stability of a graph eigenvalue with respect to a perturbation by magnetic field. In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. We will also discuss some applications of the above results.
 
The talk is based on joint works with R.Band, P.Kuchment, H.Raz, U.Smilansky and T.Weyand.