Wikibooks: Quantum Graphs/Introduction
Studying operators of Schrödinger type on metric graphs is a growing subfield of mathematical physics which is motivated both by direct applications of the graph models to physical phenomena and by use of graphs as a simpler setting in which to study complex phenomena of quantum mechanics such as An...
| Format: | Book |
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| Language: | English |
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| Online Access: | https://en.wikibooks.org/wiki/Quantum_Graphs/Introduction |
| Summary: | Studying operators of Schrödinger type on metric graphs is a growing subfield of mathematical physics which is motivated both by direct applications of the graph models to physical phenomena and by use of graphs as a simpler setting in which to study complex phenomena of quantum mechanics such as Anderson localization universality of spectral statistics nodal statistics scattering and resonances to name but a few. The name quantum graphs is most likely a shortening of the title of the article Quantum Chaos on Graphs by Kottos and Smilansky \cite{KotSmi prl97}. The model itself has been studied well before the name appeared for example in \cite{Pau jcp36 RueSch jcp53 Rot crasp83 Bel laa85 Nic incol85}. Several reviews and monographs cover various directions within the quantum graphs research \cite{GnuSmi ap06 Post book12 Mugnolo book}. However when starting a research project with students both (post ) graduate and undergraduate the author felt that a more elementary introduction would be helpful. The present manuscript grew out of the same preparatory lecture repeated at different points of time to several students. It is basically a collection of minimal examples of quantum graphs which already exhibit behavior typical to larger graphs. We supply the examples with pointers to the more general facts and theorems. Only in the last sections we explore a research topic (the nodal statistics on graphs) in some depth. For obvious reasons the pointers often lead to the monograph \cite{BerKuc graphs}. BookCat = Sources = refbegin cite journal last1=Kottos first1= Tsampikos last2= Smilansky first2= Uzy date=December 1997 title=Quantum Chaos on Graphs journal=Phys. Rev. Lett. volume= 79 pages=4794 4797 doi=10.1103/PhysRevLett.79.4794 refend2 refbegin cite journal last1= Pauling first1= Linus date=October 1936 title=The Diamagnetic Anisotropy of Aromatic Molecules journal=J. Chem. Phys. volume=4 issue=10 pages=673 677 doi=https //doi.org/10.1063/1.1749766 refend2 |
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