On the spectrum of the Laplace operator of metric graphs attached at a vertex – Spectral determinant approach

We consider a metric graph G made of two graphs G1 and G2 attached at one point. We derive a formula relating the spectral determinant of the Laplace operator SG(γ) = det(γ−∆) in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of n graphs. The formulae are also valid for the spectral determinant of the Schrödinger operator det(γ −∆+ V (x)). PACS numbers : 02.70.Hm ; 02.10.Ox Introduction.– Let us consider a bounded compact domain D1, part of a manifold. We denote by Spec(−∆;D1) the set of solutions E of −∆ψ(r) = Eψ(r) with ψ(r) satisfying given boundary conditions at the boundary ∂D1 (Sturm-Liouville problem). Similarly we consider a second bounded compact domain D2, distinct from D1 and denote Spec(−∆;D2) the spectrum of the Laplace operator in D2. Now, if we can glue D1 and D2 by identification of parts of ∂D1 and ∂D2 in order to form a unique compact domain D, the question is : can we relate the spectrum Spec(−∆;D) to Spec(−∆;D1) and Spec(−∆;D2) ? The aim of this article is to show that it is indeed possible in the particular case of metric graphs when two graphs are attached at one point. For that purpose the spectral information is encoded in the spectral determinant of the graph G, formally defined as SG(γ) = det(γ − ∆). We first define basic notations and briefly recall some results on the spectral determinant of metric graphs. We derive the relation between the spectral determinant of a graph in terms of the two subgraph determinants, when subgraphs are attached by one point, as represented on figure 2.c. The relation is generalized to describe attachment of n > 2 graphs (figure 2.d) and to deal with Schrödinger operator. Metric graphs.– Let us consider a collection of V vertices, denoted here by greek letters α, β ..., connected between each others by B bonds, denoted (αβ), (μν)... Each bond is associated with two oriented bonds, that we call arcs and denote as αβ, βα, μν, νμ.... The topology of the graph is characterized by its adjacency (or connectivity) matrix aαβ : aαβ = 1 if (αβ) is a bond and aαβ = 0 otherwise. Up to now we have built a “combinatorial graph”. If now each bond is identified with an interval [0, lαβ ] ∈ R, where lαβ is the length of the bond (αβ), the set of all connected bonds forms a “metric graph” (also called a “quantum graph”). A scalar function φ(x) living on a graph G is defined by B components φαβ(xαβ) where xαβ ∈ [0, lαβ ] is the coordinate along the bond (xαβ = 0 corresponds to vertex α and xαβ = lαβ to vertex β). By construction xαβ+xβα = lαβ. Note that components are labelled by arc variables,