Edge Switching Transformations of Quantum Graphs

Abstract

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {En} ∞ n=1 and { En} ∞ n=1 correspondingly, are level-2 interlaced, so that En−2 ≤ En ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs. [2,3], and they are defined and discussed at great length and detail in review articles and books (see e.g., [4][5][6][7]). In this note, we consider a general compact quantum graph, G = (V, E, L), whose edges e ∈ E are metrized and of finite lengths belonging to the list, may include external potential V and possibly also a magnetic potential A:The operator definition is incomplete without specifying also the boundary conditions (bc) at vertices. They are assumed here to be local, i.e., expressed in terms of linear relations on the limiting values of the function and of its derivatives along the deg(v) edges which meet at each vertex v ∈ V. The possible choices which ensure self-adjointness are reviewed in detail in, e.g., [5,6]. Under mild conditions on V and A the spectrum of the Schrödinger operator {E n } ∞ n=1 is discrete and bounded below, but not above. It suffices to assume V, A are integrable, but for a more transparent presentation we focus on the case these are piecewise continuous [6].This spectrum of H is unaffected by the operation of edge splitting, through the insertion of a vertex of degree 2 with Kirchhoff boundary conditions. These require Ψ to be continuous at v ∈ V having there directional derivatives satisfying:(To avoid confusion: these boundary conditions are not assumed here for the other quantum graph vertices.) Our purpose here is to discuss the effects on the spectrum of another basic graph transformation, that of edge switching. It is defined in the following extension of a notion which is used in combinatorial graph theory [8]. Definition 1.1 For a quantum graph G with a selfadjoint Schrödinger operator H, an edge switch is a transformation in which a pair of edges in E exchange the graph designations of one of their end points. Preserved under this exchange are the edge lengths, the local action of H along the corresponding edges, and the vertex boundary conditions -up to the corresponding transposition of the functions whose limiting behavior at the two vertices enter the local boundary conditions.Under an edge switch the collection of the edge lengths remains unchanged. However the spectra will in the generic case be affected. For example, for quantum graph Laplacians it is shown in [9] that the topology of the quantum graph with rationally independent edge length L "can be heard" in the sense of Marc Kac [10]. For clarity let us add that the graph topology may be affected by a switch, but it does not have to. However, regardless of that, and even in case the swi...

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