Spectral theory of the Schrödinger operators on fractals (Stanislav Molchanov UNC Charlotte)

Spectral properties of the Laplacian on the fractals as well as related topics (random walks on the fractal lattices, Brownian motion on the Sierpinski gasket etc.) are well understood. The next natural step is the analysis of the corresponding Schr?dinger operators and not only with random ”ergodic” potentials (Anderson type Hamiltonians) but also with the classical potentials: fast decreasing, increasing or ”periodic” (in an appropriate sense) ones. The talk will present several results in this direction. They include a) Simon – Spencer type theorem (on the absence of a.c. spectrum) and localization theorem for the fractal nested lattices (Sierpinski lattice) b) Homogenization theorem for the random walks with the periodic intensities of the jumps c) Quasi-classical asymptotics and Bargman type estimates for the Schrödinger operator with the decreasing gasket d) Bohr asymptotic formula in the case of the increasing to infinity potentials e) Random hierarchical operators, density of states and the non-Poissonian spectral statistics Some parts of the talk are based on joint research with my collaborators (Yu. Godin, A. Gordon, E. Ray, L. Zheng).