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E-book Spectral Geometry of Graphs

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Penilaian

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ifferential operators on metric graphs appear naturally in numerous applications where one is interested in describing transport or propagation of waves on a metric graph—a set of edges (bonds) joined at their end-points forming vertices (summits). Such problems became popular in the last two decades and are known nowadays as quantum graphs, despite that it is not required that the investigated system has any quantum mechanical interpretation. Thinking about quantum graphs I often imagine a set of strings joined together forming a structure reminding a spiderweb, in this picture the spectrum is just the set of its eigenfrequencies. The systems that are modelled are not necessarily locally one-dimensional, it is enough that the dynamics is essentially restricted to a set of neighbourhoods of a few low-dimensional manifolds. Imagine for example the channels in Venice or Amsterdam: if you are interested in the shortest path, then approximation as a one-dimensional network is completely sufficient. If you look how different boats miss one another in the small channels, then their shape starts to play a role. If one has enough imagination1 one may trace research on quantum graphs to the works of Klaus Ruedenberg and Charles W. Scherr [456] in the 50-ies or even of Linus Pauling in the 30-ies [425]. Research on differential operators on graph-like structures experienced a renaissance in the 80-ies pushed forward by fabrication of nano-electronic devices, and since then it is present at numerous conferences on mathematical physics, operator theory and differential equations. Quantum graphs form nowadays an attractive chapter in modern mathematical physics lying on the border between differential equations, discrete mathematics and algebraic geometry. Even other neighbouring areas of mathematics such as number theory and different types of zeta functions play a very important role in the area. What is most important is that this area is still rapidly developing with many unexpected results to surprise us. Let me just mention the recent discovery that Laplacians on metric graphs lead to explicit examples of crystalline measures, which occupied specialists in Fourier analysis for decades. Let me present here the history of my personal relation to quantum graphs without any desire to be impartial or fair. I met quantum graphs (long time before they got their name) for the first time in April 1987 when Pavel Exner and Petr Šeba came all the way from Dubna to Leningrad to visit Boris Pavlov’s group and invite us to a completely new unexplored field of mathematical physics. It appeared that these problems fitted perfectly our interests connected with exactly solvable models in quantum mechanics. In particular extension theory for symmetric operators, well developed in Leningrad, play a very important role. One should mention publications by Yossi Avron [45–47] influenced the whole later development of the field. Research in the area just started and focus was on simple examples and straightforward transfer of methods and results from the classical mathematical physics. One example of this development can be found in [205], where a one-dimensional wire is coupled to a two-dimensional plane. I still remember Robert Schrader explaining to me that such coupling contradicts physical intuition and therefore the result in [205] is remarkable. Another example are the papers [235, 236] by Nikolai Gerasimenko and Boris Pavlov solving the inverse potential problem for the star graph. It was hard to expect at that stage that somebody would lift up general questions relating geometry of graphs to spectral properties of the corresponding Schrödinger operators. Surprisingly no one was even interested in giving a rigorous mathematical definition.

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Ketersediaan

Informasi Detail

Judul Seri

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No. Panggil

530.12 KUR s

Penerbit

Germany : Springer Nature., 2024

Deskripsi Fisik

644 hlm

Bahasa

English

ISBN/ISSN

9783662678725

Klasifikasi

530.12

Tipe Isi

text

Tipe Media

computer

Tipe Pembawa

online resource

Edisi

-

Subjek

Quantum Mechanics, Quantum Theory

Info Detail Spesifik

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Pernyataan Tanggungjawab

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Versi lain/terkait

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