Spectral Analysis on Graph-like Spaces

  • Olaf Post

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

Table of contents

About this book

Introduction

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.

Keywords

35PXX ; 47A10 ; 35J25; 05C50; 34B45; 47F05; 58J50 boundary triples convergence of operators in different spaces quantum graphs and their approximations

Authors and affiliations

  • Olaf Post
    • 1
  1. 1.Department of Mathematical Sciences, Science LaboratoriesDurham UniversityDurhamUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-23840-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-23839-0
  • Online ISBN 978-3-642-23840-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book