Abstract. This paper studies the Laplacian operator on a metrized graph, and its spectral theory. 1.

Abstract. In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a “Laplacian”) with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the indextheory context by factoring the Laplacian into two first-order operators, H = A ∗ A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A ∗.

We give a review of semi-classical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semi-classical bounds to multi-particle systems, in particular, large atoms and the stability of matter.

Graphs are usually considered to be discrete objects, so issues of continuity and differentiability don’t typically appear in graph theory texts. Here is a picture of a graph:

Abstract. We define sets with finitely ramified cell structure, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates. Contents

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. A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become uniformly distributed on the metrized graph, then the ith largest eigenvalue of the Laplacian matrices of these combinatorial graphs converges to the ith largest eigenvalue of the continuous Laplacian operator on the metrized graph upon suitable scaling. The eigenvectors of these matrices can be viewed as functions on the metrized graph by linear interpolation. These interpolated functions form a normal family, any convergent subsequence of which limits to an eigenfunction of the continuous Laplacian operator on the metrized graph. Contents

Abstract. We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.

Abstract. We study spectra of alloy-type random Schrödinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloy-type model needs to have fixed sign, but the considered metric graph does not need to have a periodic structure. The second result we obtain is an exhaustion construction of the integrated density of states for ergodic random Schrödinger operators on metric graphs with a Z ν-structure. For certain models the two above results together imply the Lipschitz continuity of the integrated density of states. 1.

Abstract. The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. The necessary and sufficient condition for a Liouville type theorem to hold is that the real Fermi surface of the elliptic operator consists of finitely many points (modulo the reciprocal lattice). Thus, such a theorem generically is expected to hold at the edges of the spectrum. The precise description of the spaces of polynomially growing solutions depends upon a ‘homogenized ’ constant coefficient operator determined by the analytic structure of the dispersion relation. In most cases, simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves. The role of the base of the covering (in particular its dimension) is rather limited, while the deck group is of the most importance. The results are also established for overdetermined elliptic systems, which in particular leads to Liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs. 1.