Spectral Geometry of Graphs

Pavel Kurasov is a professor at Stockholm University. He graduated in mathematical physics with Boris Pavlov at Leningrad University and in mathematical analysis with Jan Boman at Stockholm University. He is the author of more than 100 research articles and of the book on singular interactions of differential operators together with Sergio Albeverio.

"The book provides a comprehensive introduction into the spectral theory of second-order differential operators on metric graphs by describing competing approaches. It will definitely become a standard reference source for researchers working in the area. ... Every chapter is rather short, suitable for a two-hour lecture and equipped with exercises to help understand the subject. Overall, it is easy to read the book." (Mohammed El Aïdi, Mathematical Reviews, March, 2025)

• - 1. Very Personal Introduction. - 2. How to Define Differential Operators on Metric Graphs. - 3. Vertex Conditions. - 4. Elementary Spectral Properties of Quantum Graphs. - 5. The Characteristic Equation. - 6. Standard Laplacians and Secular Polynomials. - 7. Reducibility of Secular Polynomials. - 8. The Trace Formula. - 9. Trace Formula and Inverse Problems. - 10. Arithmetic Structure of the Spectrum and Crystalline Measures. - 11. Quadratic Forms and Spectral Estimates. - 12. Spectral Gap and Dirichlet Ground State. - 13. Higher Eigenvalues and Topological Perturbations. - 14. Ambartsumian Type Theorems. - 15. Further Theorems Inspired by Ambartsumian. - 16. Magnetic Fluxes. - 17. M-Functions: Definitions and Examples. - 18. M-Functions: Properties and First Applications. - 19. Boundary Control: BC-Method. - 20. Inverse Problems for Trees. - 21. Boundary Control for Graphs with Cycles: Dismantling Graphs. - 22. Magnetic Boundary Control I: Graphs with Several Cycles. - 23. Magnetic Boundary Control II: Graphs on One Cycle and Dependent Subtrees. - 24. Discrete Graphs.