JOURNAL OF FUNCTIONAL ANALYSIS 57, 207-23 1 (1984) Scattering Theory for High Order Operators in Domains with Infinite Boundary* R. WEDER Institute de Investigaciones en Matemdticas Aplicadas y en Sistemas, Universidad National Autdnoma de M&ico. Apartado Postal 20-726. 01000 MPxico, D.F. Communicated by the Editors Received October 22, 1982; revised November 10, 1983 A general scattering theory is studied for high order elliptic operators in domains with infinite boundary. I. INTRODUCTION The spectral properties and the scattering theory for elliptic operators defined in a domain Q contained in iR, has been investigated extensively. Most of the work has been done in the case of exterior domains (i.e., domains which are the complement of a compact set). This case has been investigated by Birman [l-2], Lax and Phillips [3] (and the references quoted there), Wilcox [4, 51, Ikebe [6], Shenk and Thoe [7], Mochizuki [8], Dr?ic (91, Jtiger [lo-121, Kuroda 1131, Deift and Simon [14], Jensen and Kato [ 151, Simon [ 161, Reed and Simon [ 171, Deift [ 181, and others. The case of periodic boundary has been studied by Wilcox [19-221 and Sienz [23-251. Related problems have been studied by Wilcox (26,271, who considered stratified fluids, and by Davies and Simon [28] who studied potentials periodic in all but one direction. In the case where the domain D has nonperiodic unbounded boundary much less is known. The limiting absorption principle has been considered by Eidus and Vinnik [29], and by Eidus [30]. The existence of wave operators has been studied by Tayoshi [3 I] and Combes and Weder [32]. The completeness of wave operators in the case where the domain is asymptotic to a cone has been considered by Constantin [33]. Combes and Weder [32] proved completeness in the case where the obstacle is a surface and general boundary conditions are given, moreover it was proved that the * Research partially supported by CONACYT under Grant PCCBNAL 790025. 207 0022.1236184$3.00 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.