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.
We prove that the Dry Ten Martini Problem, i.e., all possible spectral gaps are open, holds for almost Mathieu operator with noncritical coupling and any irrational frequency. It is joint work with Artur Avila and Jiangong You.
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.
In this talk, we will talk about two phase transiton results for quasi-Periodic Schr\"odinger Operators.
For continuous Sch\"odinger operators with large analytic quasi-periodic
potentials of two frequencies, we obtain the exact power-law for phase transition in energy.
For the almost Mathieu operator with any fixed frequency, we locate
the point where phase transition from singular continuous spectrum to pure point spectrum takes place,
The key to set up Anderson Localization is to estimate the Green function. In this talk, I will introduce two ways to estimate the Green function.
One is the way of Harmonic analysis(based on Bourgain's book," Green's function estimates for lattice Schrodinger operators and application"). The other way is by Jitomirskaya (based on two papers in the Annals of Math( 1999 and 2009). In the end, I will give an
extended result on Anderson Localization (this is a joint work with Xiaoping Yuan).