On the spectrum of the hierarchical Schrödinger type operator
The goal of this paper is the spectral analysis of the Schrödinger type operator H=L+V, the perturbation of the Taibleson-Vladimirov multiplier L=𝔇α by a potential V. Assuming that V belongs to a certain class of potentials we show that the discrete part of the spectrum of H may contain negative energies, it also appears in the spectral gaps of L. We will split the spectrum of H in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential V, and low energy part which lies in the spectrum of certain bounded Schrödinger-type operator acting on the Dyson hierarchical lattice. We pay special attention to the class of sparse potentials. In this case we obtain precise spectral asymptotics for H provided the sequence of distances between locations tends to infinity fast enough. We also obtain certain results concerning localization theory for H subject to (non-ergodic) random potential V. Examples illustrate our approach.