  This paper is devoted to the spectral theory of the Schr\"{o}dinger operator
on the simplest fractal: Dyson's hierarchical lattice. An explicit description
of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided
for the unperturbed operator, i.e., for the Dyson hierarchical Laplacian.
Positive spectrum is studied for the perturbations of the hierarchical
Laplacian. Since the spectral dimension of the operator under consideration can
be an arbitrary positive number, the model allows a continuous phase transition
from recurrent to transient underlying Markov process. This transition is also
studied in the paper.
