Contact interactions on graph superlattices

Abstract

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or δ type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the δ ′ interaction and its symmetrized version for particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the δ ′ lattices have bands whose widths are asymptotically bounded and do not approach zero, while the δ lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the δ coupling constant is small enough, the δ lattice has no gaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.

A single-mode quantum transport in serial-structure geometric scatterers

“…We index the transmission and reflection amplitudes for the array by N . In analogy with (2.3) we have 16) where ε := e ikℓ . Next we add the (N +1)-th scatterer to the right side of the array for which 17) where, of course, B N + = A + and B − = A N − .…”

“…Figure 1: A loop-graph scatterer in a magnetic field only recently as a tool to describe systems of quantum wires -see [2,7,9,10,13,15,16,17,18,20,21,25,27,28,31,34] and references therein. There are also other systems for which graph description could prove to be useful such as objects composed of carbon nanotubes [14,30].…”

“…At the same time, the simplified description may still preserve basic features of the real system. Of course, replacing a branched waveguide system by its skeleton graph is a nontrivial approximation, but we shall avoid discussing this point here; some comments can be found in [16,36] and references given there.…”

“…Should the Hamiltonian be time-reversal invariant, the junction is then characterized by three real parameters. In this framework we are able to handle imperfect contacts [17]; moreover, it is straightforward to modify the results derived below to graphs with a δ ′ -coupling which corresponds to the situation where the junction itself represents a complicated geometric scatterer (see [9,16], more about that will be said in the next section). Computing the S-matrix we also assume that the particle is under influence of a potential on the stubs; this makes it possible to investigate how the band-form zones of high transmission which arise for N ≫ 1 change when an external field is applied.…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…A graph with δ ′ s couplings has important applications in lattice Kronig-Penney models, and the δ ′ s couplings at a d adge vertex can be approximated by means of d +1 couplings of the δ-type [9]. The question of physical meaning of such a coupling on graphs was addressed and a pair of simple nontrivial examples of the so-called δ ′ s couplings was presented in [12,13]. Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [20,21,22,31,39] and inverse [4,23,32,38,40,41,42], etc.…”