Degenerate band edges in periodic quantum graphs
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Degenerate band edges in periodic quantum graphs Gregory Berkolaiko1 · Minh Kha2 Received: 24 January 2020 / Revised: 17 June 2020 / Accepted: 2 July 2020 © Springer Nature B.V. 2020
Abstract Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet–Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of Z3 -periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons. Keywords Spectral theory · Mathematical physics · Quantum graphs · Periodic differential operators · Maximal abelian coverings · Band edges · Floquet–Bloch theory Mathematics Subject Classification 34B45 · 05C50 · 34L05 · 35J05 · 35P15 · 58J50
1 Introduction Periodic media play a prominent role in many fields including mathematical physics and material sciences. A classical instance is the study of crystals, one of the most stable forms of all solids that can be found throughout nature. In a perfectly ordered crystal, the atoms are placed in a periodic order and this order is responsible for many properties particular to this material. On the mathematical level, the stationary
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Minh Kha [email protected]; [email protected] Gregory Berkolaiko [email protected]
1
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
2
Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA
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G. Berkolaiko, M. Kha
Schrödinger operator −Δ + V with a periodic potential V is used to describe the one-electron model of solid-state physics [1]; here V represents the field created by the lattice of ions in the crystal. The resulting differential operator with periodic coefficients has been studied intensively in mathematics and physics literature for almost a century. A standard technique in spectral analysis of periodic operators is called the Floquet–Bloch theory (see, e.g., [26,27]). This technique is applicable not only to the above model example of periodic Schrödinger operators on Euclidean space, but also to a wide variety of elliptic periodic equations on manifolds and branching structures (graphs). Periodic elliptic operators of mathematical physics as well as their periodic elliptic counterparts on manifolds and quantum graphs do share an important feature of their spectra: the so-called band-gap structure (see, e.g., [10,24,26,27]). Namely, the spectrum of a periodic elliptic operator can be represented in a natural way as the union of finite closed intervals, called spectral bands, and sometimes they may lea
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