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 TMF, 1999, Volume 120, Number 1, Pages 99–115 (Mi tmf763)

The reduced semiclassical description method

A. A. Lobashev, N. N. Trunov

D. I. Mendeleev Institute for Metrology

Abstract: We develop a new version of the semiclassical analysis of a system of bound states in centrally symmetrical potentials. The set of potentials is in a $1:1$ correspondence with a certain set of pairs of functions of the orbital momentum. The first of these functions determines the usual WKB quantization condition and groups the potentials into equivalence classes. Its Mellin transform demonstrates similar behavior for the typical potentials, which allows describing the equivalence class with a small number of parameters. We can chose these parameters as the asymptotically exact estimates of the number of states. We obtain an equation that allows classifying states in a self-consistent atomic potential without knowing the explicit form of the potential. The second of these functions distinguishes the potentials within an equivalence class and also gives the first correction to the quantization condition.

DOI: https://doi.org/10.4213/tmf763

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English version:
Theoretical and Mathematical Physics, 1999, 120:1, 896–909

Bibliographic databases:

Citation: A. A. Lobashev, N. N. Trunov, “The reduced semiclassical description method”, TMF, 120:1 (1999), 99–115; Theoret. and Math. Phys., 120:1 (1999), 896–909

Citation in format AMSBIB
\Bibitem{LobTru99} \by A.~A.~Lobashev, N.~N.~Trunov \paper The reduced semiclassical description method \jour TMF \yr 1999 \vol 120 \issue 1 \pages 99--115 \mathnet{http://mi.mathnet.ru/tmf763} \crossref{https://doi.org/10.4213/tmf763} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1737205} \zmath{https://zbmath.org/?q=an:0966.81019} \transl \jour Theoret. and Math. Phys. \yr 1999 \vol 120 \issue 1 \pages 896--909 \crossref{https://doi.org/10.1007/BF02557399} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000083293900008} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Lobashev, N. N. Trunov, “An integral semiclassical method for calculating the spectra for centrally symmetric potentials”, Theoret. and Math. Phys., 124:3 (2000), 1250–1264
2. Tarbeev, YV, “The periodic system and the metrology of mesoscopic quantum systems”, Measurement Techniques, 44:9 (2001), 944
3. Tarbeyev Y.V., Trunov N.N., Lobashev A.A., Kukhar V.V., “Effective quantum number for centrally symmetric potentials”, Operator Methods in Ordinary and Partial Differential Equations, Operator Theory : Advances and Applications, 132, 2002, 403–410
4. K. S. Mamaeva, N. N. Trunov, “Wave Equations in Riemannian Spaces”, Theoret. and Math. Phys., 135:1 (2003), 520–530
5. N. N. Trunov, “A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved”, Theoret. and Math. Phys., 138:3 (2004), 407–417
6. Lobashev, AA, “A universal effective quantum number for centrally symmetric problems”, Journal of Physics A-Mathematical and Theoretical, 42:34 (2009), 345202
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