The discrete magnetic laplaciangeometric and spectral preorders with applications

Resumen

The discrete magnetic Laplacian: geometric and spectral preorders with applications John Stewart Fabila Carrasco 1 Abstract Graph Theory is an important part of discrete mathematics with increasingly strong links to other branches, in particular to pure and applied mathematics. Graphs are also interesting spaces on which one can do analysis. This branch of the mathematics is known as spectral graph theory. On a discrete graph, the most prominent operator is the discrete Laplacian (denoted by ∆). Such operator is the discrete analogue of the Laplacian on Riemannian manifolds. The discrete Laplacian can be generalised in a natural way to include a magnetic field which is modelled by a magnetic potential function α defined on the edges of the graph. Such operator is called the Discrete Magnetic Laplacian (DML for short) and is denoted by ∆α. The motivation for the study of the DML comes from mathematical physics, where the Schr ̈odinger operator is used to describe the dynamics of a quantum physical system. Thus, the study of the spectra of such operator is an essential topic in this field. The magnetic Laplacians is the discrete analogue of the Schr ̈odinger operator with a magnetic field. An additional motivation for the study of the magnetic Laplacian ∆α is its strong link to the topology of the underlying graph. The Laplacian ∆ is an operator defined on the set of vertices of the graph, and that depends only on the vertices adjacent to it. Thus, the Laplacian is a second-order local operator. In the case that the graph is finite, the spectrum of the magnetic Laplacian is simply the set of eigenvalues of finite multiplicity. However, when the graph is infinite, the spectrum of the magnetic Laplacian is a more complex object. In this thesis, we are interested in the spectral properties of the magnetic Laplacians. We study the spectrum of the Magnetic Laplacian under some geometrical perturbation on the graph and will be interested in the existence of spectral gaps, i.e., intervals that do not contain spectrum. The gaps may be interpreted as energy regions of the system where there is no transport. Some natural questions arise from the study of the spectrum of the Laplacian. We are interested in this dissertation in the following questions: • Question 1: What can we say about the spectrum of the Laplacian on infinite graphs? • Question 2: Does the spectrum have gaps? • Question 3: Which graphs are determined by their spectrum? Answers to Question 1 in this dissertation. We treat a special kind of infinite graphs that are known as periodic graphs or covering graph, and we consider the magnetic Laplacian with periodic potential acting on it. We prove that the spectrum of the Laplacian on the periodic graph can be reduced to the study the spectrum of the magnetic Laplacian in the finite quotient graph. Thus, information on the spectrum on the finite quotient can be lifted to the periodic graph. Answers to Question 2 in this dissertation. We study the spectral gaps of the magnetic Laplacian on periodic graphs and give a geometrical criterion to detect spectral gaps. For the study of the magnetic Laplacian, we develop a discrete bracketing technique that will be useful to localise the spectrum of the periodic graphs. Answers to Question 3 in this dissertation. We present a new geometrical construction leading to an infinite collection of families of graphs, where all the elements in each family are (finite) isospectral (weighted) graphs for the magnetic Laplacian. The parametrization of the isospectral graphs in each family is given by a number theoretic notion: the different partitions of a natural number. The answers we give to the previous questions are based on a new and fundamental tool that we developed in Chapter 2 of the thesis on which the results of the remaining chapters are based. Concretely, we present two different preorders (i.e., a reflexive and transitive relation) on the discrete graphs. The first one is based on a geometric perturbation of the graph. The second one is related to the order of the eigenvalues of the Laplacian (which are real and non-negative). We also present some applications of these preorders in other fields like, for example, in chemical graph theory. We show in Figure 1 the structure of the thesis. As mentioned above, the preorder introduced and developed in Chapter 2, will be the base of the results presented in Chapter 4-7. In conclusion, we study the spectrum/gap structure of discrete magnetic Laplacians on weighted periodic graphs. We also developed a discrete bracketing technique in the finite quotient graph and, by some perturbation on it, we obtain a localisation of its spectrum. As a result of this theory, the following summarise the essential result, techniques and important applications. • We introduce two preorders in the set of magnetic weighted graphs to study of the spectrum of the magnetic Laplacian under some geometrical perturbations. • We interpreted the magnetic potential as a Floquet parameter. • We solve (partially) the Higuchi and Shirai’s conjecture. • We give a geometrical criterion for finding spectral gaps in the spectrum of the Laplacian on periodic graphs. Figure 1: This first figure shows the interdependences of the chapters and sug- gests some pathways through the thesis. • We develop a geometrical technique producing an infinite collection of fam- ilies of graphs, where all the elements in each family are (finite) isospectral (weighted) graphs for the magnetic Laplacian. The parametrization of the isospectral graphs in each family is based on different partitions of a nat- ural number. • We also find some interesting application of the geometrical and spectral preorder in combinatorics, discrete geometry (Cheeger’s constants) and chemical graph theory.