dorsal/arxiv
View SchemaMathematical Foundations of Regular Quantum Graphs
| Authors | R. Blümel, Yu. Dabaghian, R. V. Jensen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0203126 |
| URL | https://arxiv.org/abs/quant-ph/0203126 |
Abstract
We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically and exactly, state by state, by means of periodic orbit expansions. We prove analytically that the periodic orbit series exist and converge to the correct spectral eigenvalues. We investigate the convergence properties of the periodic orbit series and prove rigorously that both conditionally convergent and absolutely convergent cases can be found. We compare the periodic orbit expansion technique with Lagrange's inversion formula. While both methods work and yield exact results, the periodic orbit expansion technique has conceptual value since all the terms in the expansion have direct physical meaning and higher order corrections are obtained according to physically obvious rules. In addition our periodic orbit expansions provide explicit analytical solutions for many classic text-book examples of quantum mechanics that previously could only be solved using graphical or numerical techniques.
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"abstract": "We define a class of quantum systems called regular quantum graphs. Although\ntheir dynamics is chaotic in the classical limit with positive topological\nentropy, the spectrum of regular quantum graphs is explicitly computable\nanalytically and exactly, state by state, by means of periodic orbit\nexpansions. We prove analytically that the periodic orbit series exist and\nconverge to the correct spectral eigenvalues. We investigate the convergence\nproperties of the periodic orbit series and prove rigorously that both\nconditionally convergent and absolutely convergent cases can be found. We\ncompare the periodic orbit expansion technique with Lagrange\u0027s inversion\nformula. While both methods work and yield exact results, the periodic orbit\nexpansion technique has conceptual value since all the terms in the expansion\nhave direct physical meaning and higher order corrections are obtained\naccording to physically obvious rules. In addition our periodic orbit\nexpansions provide explicit analytical solutions for many classic text-book\nexamples of quantum mechanics that previously could only be solved using\ngraphical or numerical techniques.",
"arxiv_id": "quant-ph/0203126",
"authors": [
"R. Bl\u00fcmel",
"Yu. Dabaghian",
"R. V. Jensen"
],
"categories": [
"quant-ph"
],
"title": "Mathematical Foundations of Regular Quantum Graphs",
"url": "https://arxiv.org/abs/quant-ph/0203126"
},
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