The article surveys the main topics, techniques, and results of the theory of periodic
operators arising in mathematical physics and other areas. Close attention is paid to …

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The
spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a …

We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time
quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was …

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall
prove the non-existence or the discreteness of embedded eigenvalues, the limiting …

This is an expository article on discrete geometric analysis based on the lectures which the
author gave at Gregynog Hall, University of Wales, as an activity of the Project “Analysis on …

In this paper we study the spectral structure of the discrete Laplacian on an infinite graph.
We show that, for a finite graph including a certain kind of a family of cycles, the spectrum of …

We establish a connection between the stability of an eigenvalue under a magnetic
perturbation and the number of zeros of the corresponding eigenfunction. Namely, we …

Given two Hilbert spaces, HH and KK, we introduce an abstract unitary operator U on HH
and its discriminant T on KK induced by a coisometry from HH to KK and a unitary involution …

The\emph {line graph} $ L (G) $ of a simple graph $ G $ is the graph whose vertices are in
one-to-one correspondence with the edges of $ G $; two vertices of $ L (G) $ are adjacent if …

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is
known that the gap intervals $(2\sqrt {2}, 3) $ and $[-3,-2) $ achieved in cubic Ramanujan …