Many-particle systems on graphs: Investigations of Schrödinger and Dirac operators on metric graphs exhibit a long-reaching history dating back even to L. Pauling reproducing spectra (diamagnetic shift) of specific molecules. Kottos and Smilansky realized that such systems, called quantum graphs, serve as an excellent model for quantum chaos. In quantum chaos, the goal is to find fingerprints of the associated classical system (integrable, chaotic flow) in the related quantum system (e.g., spectral correlations). Hul et al. succeeded an experimental realization of quantum graphs. They confirmed RMT predictions of spectral correlations of quantum graphs, and since then many other effects of one-particle quantum graphs has been experimentally tested. Concerning spectral-geometric properties, many questions have been raised for one-particle quantum graphs such as trace-formulae, isospectrality, nodal-domain count, Neumann-domain count, and spectral gap optimization. Very recently many-particle systems on metric graphs gained attention.
Scattering and heat-kernel properties of many-particle systems on graphs: For one-particle systems on graphs, the scattering and heat-kernel properties are well-understood. We started to study scattering properties and heat-kernel asymptotics for simple two-particle toy models. We want to understand the behavior of the scattering matrix in the weak and strong coupling limit for the interacting potential for more general graphs such as star graphs. The long-term goal is to connect associated characteristic quantities of those quantum objects to topological and metric properties of the many-particle quantum system. Particularly interesting is the fact that the quantum-mechanically associated configuration space depending on the particle number and due to particle-interactions it is topologically significantly more complex than the underlying metric graph staying the same for every particle number.
Geometry of eigenmodes: Since E. Chladni’s famous experiments of vibrating plates geometric properties of eigenmodes are still a hot-topic in spectral geometry. The goal is to link geometric properties of eigenmodes of a Laplacian (elliptic operator, Bilaplacian) on a domain to the corresponding eigenvalue in its spectrum. In this way, on tries to gain information of a rather laborious accessible object such as the (frequency or energy) spectrum by a more convenient evaluation process. So simplifies Courant’s celebrated nodal-domain theorem an estimate of the spectral position of an eigenvalue to a more an easy counting process of nodal-domains.
Neumann domains: To provide a novel tool linking the spectrum of a Laplacian with the topography of its eigenfunctions, S. Zelditch suggested studying its associate Morse complex. Shortly afterward McDonald and Fulling and then Band and Fajman further elaborated on the Neumann domain partition. That partition uses the gradient flow (Neumann lines) rather than the zero set (nodal lines) of eigenfunctions to partition the domain. The restricted eigenfunction to a Neumann domain possesses Neumann boundary conditions representing a nice counterpart to the nodal-domain partition which yields Dirichlet eigenfunctions. We started to investigate the spectral-position problem of Neumann domains for specific domains and eigenfunctions. Most interestingly, the topological and spectral properties of Neumann domains show a different behavior to nodal domains. Neumann domains proved to bear complementing topological information to nodal domains for quantum graphs. An exciting numerical observation made recently by us is that nodal-lines conspicuously often possess locally extremal curvature at intersection points with Neumann lines. For illustrations, here is a plot depicting a Neumann domain partition and the curvature of a selected nodal line. For a more comprehensive discussion here is a review but also containing new results of Neumann domains on graphs and manifolds.