Abstract: We investigate quantum graphs with the preferred-orientation coupling conditions suggested by Exner and Tater [1]. In particular, we are interested in the high-energy limit of their spectra. These coupling conditions violate the time-reversal symmetry, for a particular energy, the particle approaching the vertex from a given edge leaves it through the neighbouring edge (for instance, to the left of the incoming edge) and this property is cyclical. It was previously shown that the vertex scattering matrix depends on the degree of the vertex; for an odd-degree vertex, the scattering matrix converges in the high-energy limit to the identity matrix, while even-degree vertices behave differently. This behaviour affects the transport properties of these graphs.

We study two models. The first one is a finite graph consisting of edges of Platonic solids. We find that the asymptotical distribution of the eigenvalues for the octahedron graph (having even degrees of vertices) is different from the other Platonic solids (having odd degrees of vertices), for which the eigenvalues approach the spectrum of the Neumann Laplacian on an interval. The second model consists of two types of infinite lattices. For one of them, the transport at high energies is possible in the middle of the strip and is suppressed at the edges. For the other one, the transport is possible at the edge of the strip only.

The talk will be based on two papers in collaboration with P. Exner [2, 3].

[1] P. Exner, M. Tater, Quantum graphs with vertices of a preferred orientation, Phys. Lett. A 382 (2018) 283–287.
[2] P. Exner, J. Lipovsky, Spectral asymptotics of the Laplacian on Platonic solids graphs, J. Math. Phys. 60 (2019), 122101.
[3] P. Exner, J. Lipovsky, Topological bulk-edge effects in quantum graph transport, Phys. Lett. A 384 (2020), 126390.