On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 1: Time‐independent theory

摘要

AbstractLet Ω be a local perturbation of then‐dimensional domain Ω0= Ropf;n− 1× (0, π). In a previous paper8we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω0, andv·x′ ≤ 0 on δΩ, wherex′ = (x1,…,xn− 1, 0) andvis the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω0at the frequencies ω = 1,2,… ifn≤ 3. The second part of our investigation, which will appear in a subsequent paper, is d