We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width ℓ in the common boundary. We show that such a system has at least one bound state for any ℓ > 0 . We find the corresponding eigenvalues and eigenfunctions numerically using the mode-matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase-shift spacing statistics shows that in distinction to closed pseudo-integrable billiards, the present system is essentially non-chaotic. Finally, we illustrate time evolution of wave packets in the present model.

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit N → ∞ . We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. A detailed discussion is presented for a finiteperiodic "comb"; we show how the resonance poles can be computed within the Krein formula approach. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart of the resonances coming from the decoupled-surface eigenvalues such scatterers exhibit the high-energy behavior typical for the δ ′ interaction for the physically interesting couplings.