Resonances and standing waves

Recent investigations on aperiodic waves, which are generated by time-harmonic forces in waveguides, indicate a close relationship between resonances and certain time-harmonic solutions of homogeneous boundary value problems for the wave equation (“standing waves”). This paper is motivated by the observation that, in all known cases, standing waves are connected with resonances if and only if they are subject to suitable asymptotic restrictions as |x| → ∞. These asymptotic properties are used to introduce a class of “admissible” standing waves. We prove that admissible standing waves do not exist in a certain class of local perturbations Ω of the n-dimensional domain Ω₀ bounded by the hyperplanes x_n = 0 and x_n = π. This extends a result of M. Faulhaber on the absence of eigenvalues. As we shall show in a subsequent paper, absence of admissible standing waves implies absence of resonances.