Lecture Notes in Mathematics

Spectral Analysis on Graph-like Spaces

Authors: Post, Olaf

  • A thorough analysis of quantum graphs and their approximations (graph-like spaces)
  • A self-contained explanation of the tools needed (convergence of operators in different spaces, boundary triples)
  • The book is accessible for a graduate student with some knowledge in functional analysis and operators on Hilbert spaces
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  • ISBN 978-3-642-23840-6
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About this book

Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Reviews

From the reviews:

“The monograph introduces into the asymptotic analysis of graph-like spaces in the 0-thickness limit and the convergence of associated operators and related objects. … The author has succeeded to present an extensive self-contained account of a variety of results of both pure and applied character in a way to be very useful for a graduate course or seminar. It is a very successful publication in an active area of interdisciplinary research, where spectral analysis, graph-like spaces and applications interact in a beautiful manner.” (Themistocles M. Rassias, Zentralblatt MATH, Vol. 1247, 2012)

“The book represents a valuable contribution to the literature, presenting both its author’s significant contribution to this field and a broad overview of the work done by others. It will be useful to everybody interested in dynamics the quantum one in the first place on such configuration spaces and its numerous applications.” (Pavel V. Exner, Mathematical Reviews, January, 2013)


Table of contents (1 chapters)

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-642-23840-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.95
price for USA
  • ISBN 978-3-642-23839-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Spectral Analysis on Graph-like Spaces
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2039
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-23840-6
DOI
10.1007/978-3-642-23840-6
Softcover ISBN
978-3-642-23839-0
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XV, 431
Number of Illustrations and Tables
28 b/w illustrations
Topics