Bound states and scattering in quantum waveguides coupled laterally through a boundary window

Abstract

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.

Bound states in straight quantum waveguides with combined boundary conditions

“…We slightly modify for the present purpose the variational proof of the Theorem in [4], which comes out from the variational argument of [2]. The transverse ground-state wavefunction at the "tails" of our strip is…”

“…Some very simple combinations of these conditions appear due to the symmetry of special configurations in systems studied e.g. in [4], [5] and [8]. Such "combined" systems might be also of interest directly in nanoscopic physics if interphases modelled by different conditions could be realized.…”

“…[1], [2], [3], [4], [5], [6] and references therein). Their spectral properties essentially depends on the geometry of the waveguide, especially the existence of bound states induced by curvature [1], [2], [3] or by coupling of straight waveguides through windows [4], [5] were shown. The waveguides with Neumann boundary condition were also investigated in several papers (e.g.…”

Distant perturbation asymptotics in window-coupled waveguides. I. The nonthreshold case

“…The operator H being self-adjoint, its isolated eigenvalues are real, and in view of the above observation they are smaller than one; we arrange them conventionally in the ascending order counting multiplicity. Next we use bracketing again in a way analogous to [1]: we add Neumann boundaries at segments corresponding to x 1 at the endpoints of γ ± and x 2 ∈ (−d, π). In this way we get an operator estimating H from below, and since only the window parts contribute to the spectrum below one we infer by minimax that H has finitely many eigenvalues for any l > 0 and their number has a bound independent of l.…”

“…One can think of it as of an analogue of the exponential spectral shift for a pair of distant potential wells, despite the fact that the usual methods of the Schrödinger operator theory do not work here. The aim of the present paper is to study this problem in a model example of a pair of laterally coupled waveguides, or adjacent straight hard-wall strip in the plane, coupled by a pair of "windows" in the common boundary -we refer to [1], [2], [3] for a bibliography concerning such models.…”