Georgia Institute of Technology College of Sciences

The talk will present several recent results on the singular and pure point spectra for the (random or non-random) Schrӧdinger operators on the graphs or the Riemannian manifolds of the “small dimensions”. The common feature of all these results is the existence in the potential of the infinite system of the “bad conducting blocks”, for instance, the increasing potential barriers (non-percolating potentials). The central idea of such results goes to the classical paper by Simon and Spencer. The particular examples will include the random Schrӧdinger operators in the tube (or the surface of the cylinder), Sierpinski lattice etc.