AG Mathematische Physik, Prof. Alexander Bedikov (Breslau): Hierarchical Schrödinger-type operators: the case of potentials with,local singularities

Prof. Alexander Bendikov (Breslau)

Hierarchical Schrödinger-type operators: the case of potentials with local
singularities

Abstract:

The goal of this work (joint with A. Grigoryan and S. Molchanov)
is twofold. We prove that the operator H=L+V , the perturbation of the
Taibleson-Vladimirov multiplier L=D^{α} by the potential
V(x)=b‖x‖^{-α}, b≥b_{∗}, is essentially self-adjoint and
non-negative definite (the critical value b_{∗} depends on α and will
be specified in the paper). While the operator H is non-negative definite
the potential V(x) may well take negative values, e.g. b_{∗}<0 for all
0<α<1. The equation Hu=v admiits the Green function g_{H}(x,y), the
integral kernel of the operator H⁻¹. We obtain sharp lower- and upper
bounds on the ratio of the functions g_{H}(x,y) and g_{L}(x,y). Examples
illustrate our exposition.