Spectral Analysis on Graph-like Spaces
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
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2039
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Olaf Post
Spectral Analysis on Graph-Like Spaces
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Olaf Post Durham University Department of Mathematical Sciences Science Laboratories South Road Durham DH1 3LE United Kingdom [email protected]
ISBN 978-3-642-23839-0 e-ISBN 978-3-642-23840-6 DOI 10.1007/978-3-642-23840-6 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011940288 Mathematics Subject Classification (2010): 35PXX; 47A10; 35J25; 05C50; 34B45; 47F05; 58J50 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
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Preface
In this monograph, we analyse thin tubular structures, so-called “graph-like spaces”, and their natural limits, when the radius of a graph-like space tends to zero. The limit space is typically a metric graph, i.e. a graph, where each edge is associated a length, and therefore, the space turns into a one-dimensional manifold with singularities at the vertices. On both, the graph-like spaces and the metric graph, we can naturally define Laplace-like differential operators. We are interested in asymptotic properties of such operators. In particular, we show norm resolvent convergence, convergence of the spectra and resonances. Tubular structures with small radius have attracted a lot of attention in the last years. Tubular structures are frequently used in different areas such as mathematical physics to describe properties of nano-structures, in spectral geometry to provide examples with given spectral properties, or in global analysis to calculate spectral invariants. Since the underlying spaces in the thin radius limit change, and even 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. • An abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original
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