In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in Cn, which generalize hypersurfaces defined by Lee–Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.

Dissipative Schrödinger operators on metric graphs are discussed. Vertex conditions leading to maximal dissipative operators are characterised. The language of hypergraphs is introduced and used to determine possible spectral multiplicities of the self-adjoint reductions, which depends not only on the properties of the potential but on the topologic and geometric proper­ties of the metric graph. This leads to the characterisation of all operators, not possessing any self-adjoint reduction, so-called completely non-self-adjoint operators, on compact metric graphs with delta couplings at the vertices.

In this paper a new class of isospectral graphs is presented. These graphs are isospectral with respect to both the normalized Laplacian on the discrete graph and the standard differential Laplacian on the corresponding metric graph. The new class of graphs is obtained by gluing together subgraphs with the Steklov maps possessing special properties. It turns out that isospectrality is related to the degeneracy of the Steklov eigenvalues.

This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph.

The book has two central themes: the trace formula and inverse problems.

The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book.

To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions.

We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.

Several recent achievements of Sergey Naboko in spectral theory of singular differential operators and metric graphs are described. The impact of Sergey’s work on my own research career is underlined.

It is investigated how magnetic boundary control can be used to solve inverse problems for Schrödinger operators on metric graphs. Explicit examples show that such reconstruction is sometimes possible, starting from a single contact vertex in the graph.

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic chi G:=|V|-|VD|-|E|, with |VD| denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic chi G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic chi G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic chi G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815-838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic-rather than numerical-results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45-72, 2005), Helffer et al. (Ann Inst Henri Poincare Anal Non Lineaire 26:101-138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

It is proven that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not equal to zero in any non-Dirichlet vertex.