This book provides an introduction to the mathematical theory of disorder effects on quantum
spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics …

A non-backtracking walk on a graph, $ H $, is a directed path of directed edges of $ H $ such
that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length …

We prove ballistic transport of all orders, that is,∥ xme− it H ψ∥≍ tm, for the following
models: the adjacency matrix on ℤ d, the Laplace operator on ℝ d, periodic Schrödinger …

We study basic spectral features of graph Laplacians associated with a class of rooted trees
which contains all regular trees. Trees in this class can be generated by substitution …

In this paper, we develop the radial transfer matrix formalism for unitary one-channel
operators. This generalizes previous formalisms for CMV matrices and scattering zippers …

We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal
polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is …

We prove a result of delocalization for the Anderson model on the regular tree (Bethe
lattice). When the disorder is weak, it is known that large parts of the spectrum are as purely …

We establish a quantitative criterion for an operator defined on a Galton–Watson random
tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion …

We study periodic approximations of aperiodic Schr\" odinger operators on lattices in Lie
groups with dilation structure. The potentials arise through symbolic substitution systems that …

We outline some recent proofs of quantum ergodicity on large graphs and give new
applications in the context of irregular graphs. We also discuss some remaining questions …