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

Let Ω be a local perturbation of the n ‐dimensional domain Ω 0 = Ropf; n − 1 × (0, π). In a previous paper 8 we 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 ≠ Ω 0 , and v · x ′ ≤ 0 on δΩ, where x ′ = ( x 1 ,…, x n − 1 , 0) and v is 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 Ω 0 at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.