Irreducibility of the Fermi surface for planar periodic graph operators
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Irreducibility of the Fermi surface for planar periodic graph operators Wei Li1 · Stephen P. Shipman1 Received: 11 September 2019 / Revised: 20 January 2020 / Accepted: 27 June 2020 © Springer Nature B.V. 2020
Abstract We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges, (2) has positive coupling coefficients, (3) has two vertices per period. If “positive” is relaxed to “complex,” the only cases of reducible Fermi surface occur for the graph of the tetrakis square tiling, and these can be explicitly parameterized when the coupling coefficients are real. The irreducibility result applies to weighted graph Laplacians with positive weights Keywords Graph operator · Fermi surface · Floquet surface · Reducible algebraic variety · Planar graph · Graph Laplacian Mathematics Subject Classification 47A75 · 47B39 · 39A70 · 39A14 · 39A12
1 Introduction The Fermi surface (or Fermi curve) of a doubly periodic operator at an energy E is the analytic set of complex wave vectors (k1 , k2 ) admissible by the operator at that energy. Whether or not it is irreducible is important for the spectral theory of the operator because reducibility is required for the construction of embedded eigenvalues induced by a local defect [10,11,14] (except, as for graph operators, when an eigenfunction has compact support). (Ir)reducibility of the Fermi surface has been established in special situations. It is irreducible for the discrete Laplacian plus a periodic potential in all dimensions [13] (see previous proofs for two dimensions [1], [7, Ch. 4] and three dimensions [2, Theorem 2]) and for the continuous Laplacian plus a potential that is separable in a specific way in three dimensions [3, Sect. 2]. Reducibility of the Fermi surface is attained for multilayer graph operators constructed by appropriately
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Stephen P. Shipman [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, USA
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W. Li, S. P. Shipman
coupling discrete graph operators [14] or metric graph (quantum graph) operators [6,14,15]. The multilayer graphs in [6,14,15] are inherently non-planar by their very construction—they cannot be rendered in the plane without crossing edges. This led us to ask whether planarity prohibits reducibility of the Fermi surface. In this work, we address this problem by direct computation, and we are able to give a complete answer for discrete graph operators with real coefficients and two vertices per period. It turns out that planarity together with positivity of the coefficients prohibits reducibility. Our results apply to discrete weighted graph Laplacians (Theorem 2), where the positivity of the weights is a discrete version of coercivity of second-order elliptic operators. Additionally, we find that irreducibility occurs even for complex operator coefficients, including magnetic Laplacians, except for the planar graph whose faces are th
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