Criteria for the absence and existence of bounded solutions at the threshold frequency in a junction of quantum waveguides

Abstract

In the junction Ω \Omega of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [ λ † , + ∞ ) [\lambda _\dagger , +\infty ) with a positive cutoff value λ † \lambda _\dagger . Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation − Δ u = λ † u -\Delta u=\lambda _\dagger u in Ω \Omega . The first criterion is quite simple and is convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity.