Spectral Geometry of Graphs

ISBN-13: 9783662678701 / Angielski / Twarda / 2023

Pavel Kurasov

Opis

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.

The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.

Spis treści

- 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.

O autorze

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.