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 propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion excluding basis property (Proposition 6) added, unbounded time-evolution discussed, new reference

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Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.Comment: 11 pages, 3 figure

The classical Heun equation has the form Q(z) d 2 dz 2 + P(z)where Q(z) is a cubic complex polynomial, P(z) is a polynomial of degree at most 2 and V (z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes initiated the study of the set of all V (z) for which the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V (z)'s when n → ∞. We provide an explicit description of this limiting set and give a substantial amount of preliminary and additional information about it obtained using a certain technique developed by A.B.J. Kuijlaars and W. Van Assche.