Peter A. Perry

Spectral analysis of N-body Schrodinger operators *

For a large class of two body potentials, we solve two of the main problems in the spectral analysis of multiparticle quantum Hamiltonians: explicitly, we prove that the point spectrum lies in a closed countable set (and describe that set in terms of the eigenvalues of Hamiltonians of subsystems) and that there is no singular continuous spectrum. We accomplish this by extending Mourres work on three body problems to N-body problems.

The divisor of Selberg's zeta function for Kleinian groups

We compute the divisor of Selbergs zeta function for convex co-compact, torsion-free discrete groups acting on a real hyperbolic space of dimension n + 1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = nH together with the Euler characteristic of X compacti ed to a manifold with boundary. If n is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identi ed using work of Bunke and Olbrich.

The spectrum of relativistic one-electron atoms according to Bethe and Salpeter

Bethe and Salpeter introduced a relativistic equation — different from the Bethe-Salpeter equation — which describes relativistic multi-particle systems. Here we will begin some basic work concerning its mathematical structure. In particular we show self-adjointness of the one-particle operator which will be a consequence of a sharp Sobolev type inequality yielding semi-boundedness of the corresponding sesquilinear form. Moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum.

Spectral theory and mathematical physics : a festschrift in honor of Barry Simon's 60th birthday

Spectral theory and mathematical physics : a festschrift in honor of Barry Simons 60th birthday : Quantum field theory, statistical mechanics, and nonrelativistic quantum systems / Fritz Gesztesy ... [et ah], editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 76, pt. 1) Includes bibliographical references. Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math-also be made by e-mail to reprint-permission@ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

The Miura map on the line

Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrodinger operators Lq = −d2 /dx2 + q where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Korteweg-de Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura

The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory

Using techniques of stationary scattering theory for the Schrodinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds Mn arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds Mn of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.

Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

For hyperbolic Riemann surfaces of finite geometry, we study Selbergs zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of

Scattering Poles for Asymptotically Hyperbolic Manifolds

\SL(2,{\mathbb R})

A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank

is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [20] and Muller [23] to groups which are not necessarily cofinite

Miura maps and inverse scattering for the Novikov–Veselov equation

For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmons perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.