Topics in spectral theory of almost periodic operators.

Spectral theory is the branch of mathematics studying the spectrum of infinite dimensional operators. Its physical importance lies in quantum mechanics, where physical observables such as energy are represented by self-adjoint operators acting on a Hilbert space. The motion of a quantum particle is described by the Schrodinger equation, which features an operator called the Hamiltonian.

Among all linear operators, we focus on the class of discrete Schrodinger operators. These have a simple definition as the sum of a (discrete) Laplacian and a multiplication operator called the potential which act on functions on the lattice, the class is rich enough to exhibit many of the more general phenomena of spectral theory. The potentials we are interested in belong to the class of ergodic fields, which are generated using a dynamical system. A central question in the theory of disordered systems is how do the spectral properties of the operator depend on the underlying dynamical system.

When studying the spectrum, questions of interest include its structure and its type. The spectrum can be decomposed further into absolutely continuous (AC), pure point (PP) and singular continuous (SC) parts. When the operator describes the Hamiltonian of a quantum particle, the spectral type is responsible for the properties of the particle (for example, whether the medium is a conductor or an insulator). The simplest class of Schrodinger operators consists of periodic operators, which correspond to a finite dynamical system. In this case, the spectrum is purely absolutely continuous, with a band structure (a collection of intervals on the real line).

We plan to focus on the richer class of almost-periodic potentials, the simplest example of which is obtained by sampling a continuous function along the trajectory of an irrational rotation: namely, we start at a point on the circle and rotate it by an angle which is an irrational multiple of pi, this irrational constant is known as the phase. The interest in almost periodic potentials lies in its rich and exotic spectral properties. As an example, we have the almost Mathieu operator, extensively studied in the last few decades due to a combination of its innocent-looking definition (the potential is a multiplication by cosine) and the rich properties of its spectrum (the spectrum is a Cantor-type set, and there can be all three (AC, SC and PP) types of spectra). The (SC) and (PP) parts of the spectrum of almost periodic operators depend very sensitively on the Diophantine properties of the irrational phase (how well the irrational number is approximated by rational numbers). For example, the almost Mathieu operator has (SC) spectrum for very well approximated irrational phases but it has a (PP) spectrum for a set of phases which is much larger in the sense of measure.

The goal of our project is to explore the properties of almost-periodic operators beyond the well-studied case of one-dimensional irrational rotation. One of the directions of generalisation is to operators acting in a strip. Similarly to the case of one-dimensional operators, the important tool of transfer matrices is available, however, the structure is much richer, as there are several Lyapunov exponents. One of the questions that we plan to address on the first stage of the project is the length of the bands of periodic approximations of the operator. According to a plausible argument of Thouless, this should be connected to the slowest Lyapunov exponent. This has not been fully mathematically proved even for one-dimensional operators; in particular, for almost periodic operators defined by an irrational rotation, it is not clear whether this property is sensitive to the Diophantine properties of the angle. The importance of this question lies in the possible application to the study of metric properties of the spectrum (measure, Hausdorff dimension); we plan to consider such applications on further stages of the project.