Self-Adjoint Elliptic Problems in Domains with Cylindrical Ends under Weak Assumptions on the Stabilization of Coefficients

Abstract

The general self-adjoint elliptic boundary value problems are considered in a domain $G\subset \Bbb R^{n+1}$ with finitely many cylindrical ends. The coefficients are stabilizing (as $x\to\infty$, $x\in G$) so slowly that we can only describe some ``structure'' of solutions far from the origin. This problem may be understood as a model of ``generalized branching waveguide.'' We introduce a notion of the energy flow through the cross-sections of the cylindrical ends and define outgoing and incoming ``waves.'' An augmented scattering matrix is introduced. Analyzing the spectrum of this matrix one can find the number of linearly independent solutions to the homogeneous problem decreasing at infinity with a given rate. We discuss the statement of problem with so-called radiation conditions and enumerate self-adjoint extensions of the operator of the problem.