This article provides an overview of Jan Boman's illustrious seventy year career as an approximation theorist, microlocal analyst, and integral geometer. We will include his main mathematical themes and some personal observations.

Our study focuses on isoscattering quantum graphs and their eigenvalues. Contrary to the prevailing assumption that isoscattering graphs are isospectral, we present a new example of non-isospectral isoscattering graphs and study their properties. Additionally, we expanded our research to include dissipative graphs that can be realized and studied experimentally using microwave networks.

A concept of germ graphs and the 𝑀-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the 𝑀-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

The mathematical definitions of quasicrystals and crystallines are discussed within the framework of aperiodic crystals. It is shown that these are two different generalisations of ideal crystals, sharing long-range order and having completely different properties regarding short-range behaviour.

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.

Using the formalism of the Titchmarsh-Weyl M-function we study a pair of isoscattering non-isospectral quantum graphs. These graphs have the same total length L, but different topologies. The isoscattering but non-isospectral quantum graphs are difficult to identify and very complex to analyze. However, failing to recognize their existence will lead to incorrect interpretation of the spectra. We show that a potential experimental solution to this problem could involve constructing a pair of parameter-dependent non-isoscattering and non-isospectral graphs with the same total length L. Subsequently, by implementing the transition of the parameter towards zero, it can be demonstrated that the graphs transform into a pair of isoscattering graphs that remain non-isospectral. These theoretical predictions are subjected to experimental verification using microwave networks.

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.