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Differential and pseudo-differential operators on graphs as models of mesoscopic systems. (English) Zbl 1200.82086
Begehr, Heinrich G.W. (ed.) et al., Analysis and applications–ISAAC 2001. Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1384-1/hbk). Int. Soc. Anal. Appl. Comput. 10, 7-30 (2003).
Summary: Differential (rather than difference) equations on graphs have arisen during the past decade as models of many important physical systems ranging from molecules with conjugated bonds in chemistry to quantum wires, to photonic crystals. In these cases one often deals with propagation of electronic, electromagnetic, or acoustic waves in thin domains forming a graph-like structure. In the asymptotics of a very thin domain, one expects the wave propagation to be governed by a model on the resulting graph. The talk will contain a survey of this area of research. The particular topics that will be addressed include description of the situations when such models arise naturally, known and conjectured results on convergence to the graph models, and some conclusions one can derive from such models.
For the entire collection see [Zbl 1031.35002].

82D99 Applications of statistical mechanics to specific types of physical systems
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
81Q99 General mathematical topics and methods in quantum theory
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics