Infinite Quantum Graphs

The development of classical spectral theory (theory of self-adjoint operators in a Hilbert space) is related to Quantum Mechanics: a phase space is a Hilbert space and each observable is given by a linear self-adjoint operator acting in this space. The cornerstone of classical spectral theory is the Spectral Theorem, which allows to distinguish certain simplest "harmonics" of a self-adjoint operator and then to construct the evolution induced by its action. During the last two decades, quantum graphs became an extremely popular subject due to their numerous applications in mathematical physics, chemistry, nanotechnology and engineering. Moreover, they are a good laboratory to study complicated properties of quantum systems. Quantum graphs exhibit mixed dimensional properties being locally one-dimensional but globally multi-dimensional of many different types. The standard restriction on the geometry of underlying metric graphs is the existence of a positive lower bound for the length of its edges. This restriction clearly excludes a number of interesting models. Moreover, in this case graphs are close to fractals in their geometric properties. The main aim of our project is the spectral analysis of quantum graphs without this standard restriction.

During the last two decades, quantum graphs (Laplacians or Schrödinger-type operators on metric graphs) became an extremely popular subject due to their numerous applications in mathematical physics, chemistry, nanotechnology and engineering. Moreover, they are a good laboratory to study complicated properties of quantum systems. Metric graphs exhibit mixed dimensional properties being locally one-dimensional but globally multi-dimensional of many different types. Moreover, quantum graphs play an important role as an intermediate setting between Laplacians on Riemannian manifolds and Laplacians on graphs and share properties with both. The most common restriction on the geometry of underlying metric graphs is the existence of a positive lower bound for the length of its edges. This restriction clearly excludes a number of interesting models and phenomena. The main aim of our project is the analysis of quantum graphs without this standard restriction. As our key result, we established a close relationship between main spectral properties of Kirchhoff Laplacians on metric graphs and a certain class of weighted graph Laplacians, which, however, are in general unbounded if the underlying metric graph admits arbitrarily short edges. This connection turned out to be very fruitful. On the one hand, the theory of weighted graph Laplacians have seen a tremendous progress during the last decade and we exploited these results to investigate Laplacians on metric graphs. In turn, in the framework of Dirichlet forms, Laplacians on metric graphs give rise to (strongly) local Dirichlet forms. Thus, the established connections provide us with new proofs and another perspective on the recent progress in weighted graph Laplacians.