The discrete Klein-Gordon equation on a two-dimensional square lattice satisfies an ℓ 1 → ℓ ∞ dispersive bound with polynomial decay rate |t| −3/4 . We determine the shape of the light cone for any choice of the mass parameter and relative propagation speeds along the two coordinate axes. Fundamental solutions experience the least dispersion along four caustic lines interior to the light cone rather than along its boundary, and decay exponentially to arbitrary order outside the light cone. The overall geometry of the propagation pattern and its associated dispersive bounds are independent of the particular choice of parameters. In particular there is no bifurcation of the number or type of caustics that are present.The discrete Klein-Gordon equation is a classical analogue of the quantum harmonic lattice. In the quantum setting, commutators of time-shifted observables experience the same decay rates as the corresponding Klein-Gordon solutions, which depend in turn on the relative location of the observables' support sets.

We consider a large class of harmonic systems, each defined as a quasi-free dynamics on the Weyl algebra over ℓ 2 (Z d ). In contrast to recently obtained, short-time locality estimates, known as Lieb-Robinson bounds, we prove a number of long-time dispersive estimates for these models.

This dissertation is concerned with various aspects of the spectral theory of differential and pseudodifferential operators. It consists of two chapters.

ABSTRACT. We discuss convergence properties of the spectral shift functions associated with a pair of Schrödinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches infinity.

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schrödinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite.