Vertex Couplings in Quantum Graphs: Approximations by Scaled Schrödinger Operators

Abstract

We review recent progress in understanding the physical meaning of quantum graph models through analysis of their vertex coupling approximations.

Dispersion for the Schrödinger equation on networks

“…with ψ λ (x), β λ ∈ R, I λ ∈ {I e } e∈E and λ∈R |b λ | < ∞. Then decomposition (6) is implied by (12), (13) and the fact that for t > 0 and r ∈ R…”

“…We mention some of them: carbon nano-structures [26], photonic crystals [14], high-temperature granular superconductors [1], quantum waveguides [8], free-electron theory of conjugated molecules in chemistry, quantum chaos, etc. For more details we refer the reader to review papers [23], [24], [17] and [13].…”

Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains

“…When all the constants α j are equal the operator defined by the choice (1.1) is called Kirchhoff (or standard) Laplacian and the boundary conditions read x 1 (0) = ... = x N (0) and N i=1 x ′ i (0) = 0. A central problem in using operators in the family −∆ Π,Θ G to approximate the dynamics generated by the Laplacian in a network of thin tubes squeezing to a graph is to understand which boundary conditions in the vertex arise in limit (see, e.g., [17,18,38] for a review on this topic). It turns out that this problem strongly depends on what kind of Laplacian is taken in the squeezing network.…”

An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions

“…Define the "reduced response operator" on e j , with j ≥ 2, by R 0,j p (t) =ũ p j (0, t) associated to the IBVP ( 17)- (20). From (21), we immediately obtain Lemma 1.…”

“…Here, T is arbitrary positive number, q j ∈ C([a 2j−1 , a 2j ]) for all j, and f ∈ L 2 (0, T). The physical interpretation of conditions (3) and (4), and some other matching conditions was discussed in [20].…”