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.
General
| Imprint: |
Springer-Verlag
|
| Country of origin: |
Germany |
| Series: |
Operator Theory: Advances and Applications, 293 |
| Release date: |
October 2023 |
| First published: |
2023 |
| Authors: |
Pavel Kurasov
|
| Dimensions: |
235 x 155mm (L x W) |
| Pages: |
490 |
| Edition: |
1st ed. 2023 |
| ISBN-13: |
978-3-662-67870-1 |
| Categories: |
Books
|
| LSN: |
3-662-67870-5 |
| Barcode: |
9783662678701 |
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