Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the spectrum of the hierarchical Schr\"{o}dinger type operators   The goal of this paper is the spectral analysis of the Schr\"{o}dinger type
operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier
$L=\mathfrak{D}^{\alpha}$ 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 Schr\"{o}dinger-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.

36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph