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Mathematics - Analysis | Spectral Analysis on Graph-like Spaces

Spectral Analysis on Graph-like Spaces

Series: Lecture Notes in Mathematics, Vol. 2039

Post, Olaf

2012, XV, 431p. 28 illus..

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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
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.

Content Level » Research

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

Related subjects » Analysis - Dynamical Systems & Differential Equations

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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