Abstract:
We give the global and Strichartz estimates for the Schro ̈dinger maximal operators, end point maximal and the local smoothing estimates for Schro ̈dinger equation. The singular continuous and pure point spectrum of self-adjoint extensions and Laplaceians of fractul graphs are shown with the spectral Localization in the hierarchical Anderson model. The radial positive definite function with bases of subspaces, property of x-positive definiteness, general Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities are investigated. The space time estimates and the negative spectrum of the three dimentional hierarchical Schro ̈dinger operaters with pure point spectrum interactions are discussed.
