This paper is a brief review of the program initiated some years ago in [JMP]. The main goal of this program is to understand the formation and the propagation properties of surface states in regions with corrugated boundaries. The history of this problem and its physical aspects are discussed in [JMP, KP]. The results described in this review are obtained in collaborations with Yoram Last and ...

We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we ...

During the past dozen years, a major focus of my research has been the spectral theory of orthogonal polynomials—both orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC). There has been a flowering of the subject in part because of a cross-fertilization of two communities of researchers. I will discuss some aspects of this subject here; for a lot more, see my recent boo...

Contents Chapter 1. Eigenvalues and the Laplacian of a graph 1 1.1. Introduction 1 1.2. The Laplacian and eigenvalues 2 1.3. Basic facts about the spectrum of a graph 6

We introduce and study tropical eigenpairs of tensors, a generalization of the tropical spectral theory of matrices. We show the existence and uniqueness of an eigenvalue. We associate to a tensor a directed hypergraph and define a new type of cycle on a hypergraph, which we call an H-cycle. The eigenvalue of a tensor turns out to be equal to the minimal normalized weighted length of H-cycles o...

We write M ∈ Rn×n to denote that M is an n×n matrix with real elements, and v ∈ Rn to denote that v is a vector of length n. Vectors are usually taken to be column vectors unless otherwise specified. Recall that a real matrix M ∈ Rn×n represents a linear operator from Rn to Rn. In other words, given any vector v ∈ Rn, we think of M as a function which maps it to another vector w ∈ Rn, namely th...

1. Recall the spectrum of a bounded linear operator T ∈ B(X) on a Banach space X, more generally a Banach algebra element a ∈ A: 1.1 σ(a) = σA(a) = {λ ∈ C : a− λ 6∈ A−1}, where of course 1.2 a ∈ A−1 ⇐⇒ ∃a′, a′′ ∈ A, a′a = 1 = aa′′ : for example for square matrices A = Cn×n we have the well worn cliche 1.3 σ(a) = σA(a) = {λ ∈ C : det(a− λ) = 0}; for continuous functions A = C(Ω) on compact Hausd...

We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S , with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T . The zeroth law of thermodynamics postulates that, independently of its initial configuration...