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Spectral Analysis on Graph-like Spaces

  • Book
  • © 2012

Overview

Authors:
  1. Olaf Post

    1. Department of Mathematical Sciences, Science Laboratories, Durham University, Durham, United Kingdom

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  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2039)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    • Olaf Post
    Pages 1-56

  3. Graphs and Associated Laplacians

    • Olaf Post
    Pages 57-96

  4. The Functional Analytic Part: Scales of Hilbert Spaces and Boundary Triples

    • Olaf Post
    Pages 97-185

  5. The Functional Analytic Part: Two Operators in Different Hilbert Spaces

    • Olaf Post
    Pages 187-257

  6. Manifolds, Tubular Neighbourhoods and Their Perturbations

    • Olaf Post
    Pages 259-289

  7. Plumber’s Shop: Estimates for Star Graphs and Related Spaces

    • Olaf Post
    Pages 291-366

  8. Global Convergence Results

    • Olaf Post
    Pages 367-388

  9. Back Matter

    Pages 389-431
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Keywords

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)

Authors and Affiliations

  • Department of Mathematical Sciences, Science Laboratories, Durham University, Durham, United Kingdom

    Olaf Post

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