Spectral analysis on graph-like spaces
by Post, Olaf
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 limi…
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- Full title :Spectral analysis on graph-like spaces / Olaf Post.
Summary
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.
Read More ↓Edition Statements
- Publication Date:©2012.
Publication Statements
- Publisher(s):Springer
- Place of Publication:Heidelberg
- Language:eng
- Publication Date:©2012.
The Physical Object
- Description:1 online resource (xv, 431 p.) : : illustrations
Edition Identifiers
- ISBN:9783642238406, 3642238408
- oclc:ocn805298892
- LC Classification:QA689
- Dewey Decimal:515/.782
Table of Contents
1 Introduction -- 2 Graphs and associated Laplacians -- 3 Scales of Hilbert space and boundary triples -- 4 Two operators in different Hilbert spaces -- 5 Manifolds, tubular neighbourhoods and their perturbations -- 6 Plumber's shop: Estimates for star graphs and related spaces -- 7 Global convergence results.
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