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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Quantum counterparts of VIIα, IIIα=1, VIα≠1 over the harmonic oscillator in semiclassical approximation; pp. 347–354

(Full article in PDF format) doi: 10.3176/proc.2010.4.14


Authors

Eugen Paal, Jüri Virkepu

Abstract

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some real 3-dimensional Lie algebras. The Jacobi operators of these quantum algebras are studied in semiclassical approximation.

Keywords

operad, 3d real Lie algebras, operadic Lax equation, Jacobi operator

References

  1. Gerstenhaber , M. The cohomology structure of an associative ring. Ann. Math. , 1963 , 78 , 267–288.
doi:10.2307/1970343

  2. Landau , L. and Lifshitz , E. Theoretical Physics , Vol. 2: Field Theory}. Nauka , Moskva , 1973 (in Russian).

  3. Lax , P. D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. , 1968 , 21 , 467–490.
doi:10.1002/cpa.3160210503

  4. Paal , E. Invitation to operadic dynamics. J. Gen. Lie Theory Appl. , 2007 , 1 , 57–63.

  5. Paal , E. and Virkepu , J. Note on operadic harmonic oscillator. Rep. Math. Phys. , 2008 , 61 , 207–212.
doi:10.1016/S0034-4877(08)80008-0

  6. Paal , E. and Virkepu , J. 2D binary operadic Lax representation for harmonic oscillator. In Noncommutative Structures in Mathematics and Physics (Caenepeel , S. , Fuchs , J. , Gutt , S. , Schweigert , C. , Stolin , A. , and Van Oystaeyen , F. , eds). K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB) , Brussels , 2010 , 209–216.

  7. Paal , E. and Virkepu , J. Operadic representations of harmonic oscillator in some 3d Lie algebras. J. Gen. Lie Theory Appl. , 2009 , 3 , 53–59.
doi:10.4303/jglta/S090104

  8. Paal , E. and Virkepu , J. Dynamical deformations of 3d Lie algebras in Bianchi classification over harmonic oscillator. J. Math. Phys. , 2009 , 50 , 053523.
doi:10.1063/1.3131615

  9. Paal , E. and Virkepu , J. Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator. Centr. Eur. J. Phys. , 2010 , 8 , 289–295.
doi:10.2478/s11534-009-0123-8
 
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Current Issue: Vol. 67, Issue 4 in Press, 2018




Publishing schedule:
No. 1: 20 March
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