Here you can find some information on the research project P28807 Infinite Quantum Graphs funded by the Austrian Science Fund (FWF).

During the last two decades, quantum graphs became an extremely popular subject due to their numerous applic ations 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 global ly 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.

The project started in September 2016 and finished in September 2020.

My invited address at the 8th European Congress of Mathematics: "Laplacians on Infinite Graphs"

• Jonathan Eckhardt (Loughborough University, UK and University of Vienna, Austria),
• Pavel Exner (Nuclear Physics Institute, Czech Republic),
• Fritz Gesztesy (Baylor University, USA),
• Mark Malamud (RUDN, Russia),
• Delio Mugnolo (FernUni Hagen, Germany),
• Hagen Neidhardt (WIAS, Germany),
• Noema Nicolussi (University of Vienna, Austria),
• Gerald Teschl (University of Vienna, Austria)

1. Laplacians on infinite graphs: discrete vs. continuous, (with N. Nicolussi), extended version of an invited lecture at the 8ECM, June 2021 (arXiv:2110.03566/a>).
2. Heat kernels of the discrete Laguerre operators, Lett. Math. Phys. 111, 2:32 (2021) (arXiv:2007.14963)
3. A note on the Gaffney Laplacian on infinite metric graphs, (with N. Nicolussi), J. Funct. Anal. 280, no.10, Art: 109216 (2021) (arXiv:2108.08185)
4. Self-adjoint and Markovian extensions of infinite quantum graphs, (with D. Mugnolo and N. Nicolussi), J. London Math. Soc. 105, no.2, 1262-1313 (2022) (arXiv:1911.04735)
5. On the absolutely continuous spectrum of generalized indefinite strings, (with J. Eckhardt), Ann. Henri Poincaré, 22, no.11, 3529-3564 (2021) (arXiv:1902.07898)
6. On the absolutely continuous spectrum of generalized indefinite strings II, (with J. Eckhardt and T. Kukuljan), Israel J. Math., to appear (arXiv:1906.05106)
7. Quantum graphs on radially symmetric antitrees, (with N. Nicolussi), J. Spectral Theory 11, no.1 (2021) (arXiv:1901.05404)
8. The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation, (with J. Eckhardt), Int. Math. Res. Notices (IMRN), 2020, no. 16, 5126-5151 (2020) (arXiv:1801.04612)
9. Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow, (with J. Eckhardt and N. Nicolussi), J. Differential Equations 268, no. 6, 3016-3034 (2020) (arXiv:1907.01911)
10. Spectral estimates for infinite quantum graphs, (with N. Nicolussi), Calc. Var. Partial Differential Equations 58, no. 1, Art ID 15 (2019) (arXiv:1711.02428)
11. The classical moment problem and generalized indefinite strings, (with J. Eckhardt), Integr. Equat. Oper. Theory 90, 2:23 (2018) (arXiv:1707.08394)
12. On the Hamiltonian-Krein Index for a non-self-adjoint spectral problem, (with N. Nicolussi), Proc. Amer. Math. Soc. 146, no. 9, 3907-3921 (2018) (arXiv:1712.01702)
13. Spectral theory of infinite quantum graphs, (with P. Exner, M. Malamud, and H. Neidhardt), Ann. Henri Poincaré 19, no. 11, 3457-3 (2018) (arXiv:1705.01831)
14. Infinite quantum graphs, (with P. Exner, M. Malamud, and H. Neidhardt), Doklady Math. 95, no.1, 31-36 (2017)

Monograph:

• Laplacians on infinite graphs, (with N. Nicolussi), Mem. Eur. Math. Soc., to appear (PDF).

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