Ralph S. Phillips

Extremals of determinants of Laplacians

On etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

The counting numbers for discrete subgroups of motions in Euclidean and non-Euclidean spaces are obtained using the wave equation as the principal tool. In dimensions 2 and 3 the error estimates are close to the best known.

Explicit expanders and the Ramanujan conjectures

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Scattering Theory for Dissipative Hyperbolic Systems

An abstract theory of scattering is developed for dissipative hyperbolic systems, a typical example of which is the wave equation: utt = Δu in an exterior domain with lossy boundary conditions: un ∓ αut = 0, α ≥ 0. In this theory, as in an earlier theory developed by the authors for conservative systems, a central role is played by two distinguished subspaces of data common to both the perturbed and unperturbed problems. Associated with each sub-space is a translation representation of the unperturbed system. When these representations coincide they provide a convenient tool for extending the data so as to include a large class of generalized eigenfunctions for both the perturbed and unperturbed generators. The scattering matrix is characterized in terms of these generalized eigenfunctions; it is shown to be meromorphic in the whole complex plane and holomorphic in the lower half plane. The zeros and poles of the scattering matrix correspond, respectively, to incoming and outgoing generalized eigenfunctions of the perturbed generator.

The wave equation in exterior domains

This note deals with solutions of the wave equation in three dimensions in the exterior of a finite number of smooth obstacles, on whose boundaries the solution is subject to boundary conditions of the form u = 0 or un = <ru, a a non-negative function. We shall show that every such solution of finite energy propagates eventually out to infinity and behaves asymptotically like a free space solution.

Energy decay for the neutrino equation in the exterior of a torus

Local energy decay is established for the solutions of the neutrino equation in the exterior G of a torus for a class of boundary conditions, described as follows: To each energy conserving boundary condition at a point x on ∂G there corresponds a vector in the tangent plane to ∂G at x. The result has been proved when the torus and boundary conditions are axially symmetric and when the paths generated by this vector field are closed. What is novel about this problem is the fact that the boundary conditions are nowhere coercive.

Scattering theory for the wave equation on a hyperbolic manifold

This talk is a survey of work done jointly, mainly with Peter Lax [3,4,5] but also with Bettina Wilkott and Alex Woo [10], over the past ten years. It deals with the spectrum of the perturbed (and unperturbed) Laplace-Beltrami operator acting on automorphic functions on an n-dimensional hyperbolic space IHn. The associated discrete subgroup Γ of motions is assumed to have the finite geometric property, but is otherwise unrestricted. This means that the fundamental domain, when derived by the polygonal method, has a finite number of sides; its volume may be finite or infinite and it can have cusps of arbitrary rank. With the obvious identifications the fundamental domain can be treated as a manifold, M.

The Spectrum of the Laplacian for Domains in Hyperbolic Space and Limit Sets of Kleinian Groups

This is a report on a joint paper with Peter Sarnak [5] on the properties of the spectrum of the Laplacian, with free boundary conditions, for domains of infinite volume in hyperbolic space. We are mainly concerned with the discrete part of the spectrum, its existence or nonexistence, and lower bounds for the bottom of the spectrum. Combining these results with those of Patterson [4] and Sullivan [6] yields new estimates for the Hausdorff dimension of the limit sets of Kleinian groups.

Scattering theory for twisted automorphic functions

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group r with an irreducible unitary representation p and satisfying u(yz) = p(Qy)u(z). The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, R_ and R+, for the solution operator. The scattering operator, which maps R f into R+f, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of p is one, the elements of the scattering operator cannot vanish. However when dim(p) > 1 this is no longer the case.

The translation representation theorem

Short of a new theorem on semigroups of operators, a new proof of an old theorem on this subject is a suitable offering to Einar Hille on his 85th birthday.