We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.

Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

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The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos–Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scale-invariant boundary conditions the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general (frequency-dependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be “indexed ” a posteriori by truly periodic orbits. For the scale-invariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula.

Abstract. We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy. 1.

Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of the spectral density in terms of sums over periodic orbits. Finally, a general trace formula for any self adjoint realisation of the Laplacian on a compact, metric graph is given.

Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

Abstract. The aim of the present paper is to analyse the spectrum of Laplace operators on graphs. Motivated by the general form of vertex conditions of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs using the theory of boundary triples. In particular, we derive a spectral relation for equilateral metric graphs and index formulas. Moreover, we introduce extended metric graphs occuring naturally as limits of “thick ” graphs, and provide spectral analysis of natural Laplacians on such spaces. 1.

the expansion of the resolvent for elliptic boundary contact problems

Abstract. We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators. 1.