Finding the hole in a wall
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Finding the hole in a wall M. Archibald1 · S. Currie1 · M. Nowaczyk2 Received: 30 July 2020 / Accepted: 10 September 2020 / Published online: 26 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we model a graphene nano-ribbon structure by analysing an infinite 3-regular hexagonal grid which is transformed to a rectangular coordinate system or “wall”. Our goal is to solve the inverse problem of identifying the position of a single vacancy break using the lengths of the closed paths along the edges of the underlying graph. We provide an algorithm to determine the exact position of the defect by using data from at most three reference points. Keywords Quantum graph · Periodic orbit · Single vacancy break · Graphene · Inverse problem Mathematics Subject Classification 82D77 · 47N60 · 81Q10 · 34A55 · 34B45
1 Introduction Recently there have been many developments in the field of quantum graphs, in particular, using graphs to model physical structures, see [7] amongst others. In for example [4, 5, 9], we see that the topic has attracted scientists from areas such as mathematics, physics and chemistry. Many authors have considered direct and inverse spectral problems on quantum graphs. For example in [6, 8] the so-called trace formula for the Laplace operator with energy-independent matching conditions has been investigated. This formula provides a strong connection between * S. Currie [email protected] M. Archibald [email protected] M. Nowaczyk [email protected] 1
School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS, Johannesburg 2050, South Africa
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Faculty of Applied Mathematics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30‑059 Krakow, Poland
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Vol.:(0123456789)
2314
Journal of Mathematical Chemistry (2020) 58:2313–2323
the eigenvalues of the operator with the lengths of the closed paths in the graph. This phenomenon of discrete spectrum has been observed in the studies of nanostructures see [1–3]. However, experimental data shows that the spectrum of a given structure does not always produce the results predicted by the mathematical model due to some naturally occurring defects in the structure. In this paper we analyse an infinite 3-regular (each vertex has degree/valency 3) hexagonal grid which models a graphene nano-ribbon (GNR) channel region. The carbon atoms are represented by the vertices and the bonds by edges each of length one. Low energy free electrons “move” in this structure obeying the Laplace equation with so-called natural (Kirchhoff) boundary conditions at the vertices. By sending a signal from one of the vertices and detecting the returning impulses one can experimentally observe the spectrum of this structure. Since the conditions for the trace formula are fulfilled (see [6, 8]) and having experimental data of all the eigenvalues, we are able to determine the lengths of all closed paths (periodic orbits) starting and ending at the given vertex where the detector is placed. Bec
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