On the spectrum of the hierarchical Schr\"{o}dinger operator

The goal of this paper is the spectral analysis of the Schr\"{o}dinger operator $H=L+V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=\mathcal{D}^{\alpha}$ by a potential $V$. Assuming that $V$ belonges to a class of fast decreasing 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 Schr\"{o}dinger operator acting on the Dyson hierarchical lattice. The spectral asymptotics \ strictly depend on the transience versus recurrence properties of the underlying hierarchical random walk. In the transient case we will prove results in spirit of CLR theory, for the recurrent case we will provide Bargmann's type asymptotics.