Abstract:
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:
The main objective of the present work is to study contraction semigroups generated by Laplace operators on metric graphs, which are not necessarily self-adjoint. We prove criteria for such semigroups to be continuity and positivity preserving. Also we provide a characterization of generators of Feller semigroups on metric graphs.

Abstract:
The grand potential of a system of interacting electrons is considered as a stationary point of a self-energy functional. It is shown that a rigorous evaluation of the functional is possible for self-energies that are representable within a certain reference system. The variational scheme allows to construct new non-perturbative and thermodynamically consistent approximations. Numerical results illustrate the practicability of the method.

Abstract:
Self-energy-functional theory is a formal framework which allows to derive non-perturbative and thermodynamically consistent approximations for lattice models of strongly correlated electrons from a general dynamical variational principle. The construction of the self-energy functional and the corresponding variational principle is developed within the path-integral formalism. Different cluster mean-field approximations, like the variational cluster approximation and cluster extensions of dynamical mean-field theory are derived in this context and their mutual relationship and internal consistency are discussed.

Abstract:
Various types of metal-insulator transitions are discussed to find conditions for which an ideal surface of a bulk insulator is metallic. It is argued that for the correlation-driven Mott metal-insulator transition the surface phase diagram should be expected to have the same topology as the phase diagram for magnetic order at surfaces: The corresponding linearized mean-field descriptions, a simplified dynamical mean-field theory of the Hubbard model and the Weiss mean-field theory for the Ising model, are found to be formally equivalent. A new kind of surface state appears in the low-energy part of the one-particle excitation spectrum as a precursor effect of the Mott transition.

Abstract:
The basic theory of photoemission, inverse photoemission, Auger-electron and appearance-potential spectroscopy is developed within a unified framework starting from Fermi's golden rule. The spin-resolved and temperature-dependent appearance-potential spectroscopy of band-ferromagnetic transition metals is studied in detail. It is shown that the consideration of electron correlations and orbitally resolved transition-matrix elements is essential for a quantitative agreement with experiments for Ni.

Abstract:
It is shown that a minimum realization of the dynamical mean-field theory (DMFT) can be achieved by mapping a correlated lattice model onto an impurity model in which the impurity is coupled to an uncorrelated bath that consists of a single site only. The two-site impurity model can be solved exactly. The mapping is approximate. The self-consistency conditions are constructed in a way that the resulting ``two-site DMFT'' reduces to the previously discussed linearized DMFT for the Mott transition. It is demonstrated that a reasonable description of the mean-field physics is possible with a minimum computational effort. This qualifies the simple two-site DMFT for a systematic study of more complex lattice models which cannot be treated by the full DMFT in a feasible way. To show the strengths and limitations of the new approach, the single-band Hubbard model is investigated in detail. The predictions of the two-site DMFT are compared with results of the full DMFT. Internal consistency checks are performed which concern the Luttinger sum rule, other Fermi-liquid relations and thermodynamic consistency.

Abstract:
For a system of correlated electrons, the Luttinger-Ward functional provides a link between static thermodynamic quantities on the one hand and single-particle excitations on the other. The functional is useful to derive several general properties of the system as well as for the formulation of thermodynamically consistent approximations. Its original construction, however, is perturbative as it is based on the weak-coupling skeleton-diagram expansion. Here, it is shown that the Luttinger-Ward functional can be derived within a general functional-integral approach. This alternative and non-perturbative approach stresses the fact that the Luttinger-Ward functional is universal for a large class of models.

Abstract:
A pedagogical introduction to the cluster-perturbation theory, the variational cluster approximation and to self-energy-functional theory is given. Some standard applications and the relation to dynamical mean-field theory are discussed.