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Fractional Calculus and Applied Analysis

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Reflection symmetric Erdélyi-Kober type operators — A quasi-particle interpretation

Richard Herrmann
Published Online: 2014-09-03 | DOI: https://doi.org/10.2478/s13540-014-0221-1

Abstract

The reflection symmetric Erdélyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically. A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogously to the classical approach for interacting multi-particle systems, the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.

MSC: 26A33; Secondary: 70Hxx, 37Kxx, 81Q35, 81Q60

Keywords: generalized fractional calculus; shifted Riesz integrals; Erdélyi-Kober integrals; fractional operators; Fock space; quasiparticle; pairing-Hamiltonian

  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publications, New York (1965). Google Scholar

  • [2] Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory. Phys. Rev. Lett. 115 (1959), 485–491. Google Scholar

  • [3] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity. Phys. Rev. 108 (1957), 1175–1204. http://dx.doi.org/10.1103/PhysRev.108.1175CrossrefGoogle Scholar

  • [4] S. T. Belyaev, Effect to Pairing Correlations on Nuclear Properties. E. Munksgaard, Kobenhavn (1959), 55 pages; Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959). Google Scholar

  • [5] N. N. Bogoliubov and S. V. Tjablikov, In: Soviet J. Eks. Teor. Fiz. 19 (1949), 256. Google Scholar

  • [6] L. N. Cooper, Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104 (1956), 1189–1190. http://dx.doi.org/10.1103/PhysRev.104.1189CrossrefGoogle Scholar

  • [7] P. Debye, Näherungsformeln für die Zylinderfunktionen für großeWerte des Arguments und unbeschränkt veränderliche Werte des Index. Mathematische Annalen 67 (1909), 535–558; doi:10.1007/BF01450097; English transl. in P. J. W. Debye, The collected papers of Peter J.W. Debye, Interscience Publishers, Inc., New York (1954). http://dx.doi.org/10.1007/BF01450097CrossrefGoogle Scholar

  • [8] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 (1935), 777–780; doi:10.1103/PhysRev.47.777. http://dx.doi.org/10.1103/PhysRev.47.777CrossrefGoogle Scholar

  • [9] A. Erdélyi, On fractional integration and its application to the theory of Hankel transforms. The Quarterly J. of Mathematics (Oxford), Second Ser., 11 (1940), 293–303. http://dx.doi.org/10.1093/qmath/os-11.1.293CrossrefGoogle Scholar

  • [10] W. Feller, On a generalization of Marcel Riesz’ potentials and the semigroups generated by them. Comm. Sem. Mathem. Universite de Lund (1952), 72–81. Google Scholar

  • [11] W. Greiner and J. A. Maruhn, Nuclear Models. Springer, Heidelberg, Berlin — New York (1996). http://dx.doi.org/10.1007/978-3-642-60970-1CrossrefGoogle Scholar

  • [12] R. Herrmann, Fractional Calculus — An Introduction for Physicists, 2nd ed., World Scientific Publ., Singapore (2014). http://dx.doi.org/10.1142/8934CrossrefGoogle Scholar

  • [13] R. Herrmann, Towards a geometric interpretation of generalized fractional integrals — Erdélyi-Kober type integrals on R N, as an example. Fract. Calc. Appl. Anal. 17, No 2 (2014), 361–370; DOI: 10.2478/s13540-014-0174-4; http://link.springer.com/article/10.2478/s13540-014-0174-4. http://dx.doi.org/10.2478/s13540-014-0174-4CrossrefGoogle Scholar

  • [14] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ., Singapore (2000). http://dx.doi.org/10.1142/9789812817747CrossrefGoogle Scholar

  • [15] V. S. Kiryakova, Generalized Fractional Calculus and Applications. Longman (Pitman Res. Notes in Math. Ser. 301), Harlow; Co-publ.: John Wiley and Sons, New York (1994). Google Scholar

  • [16] C. Kittel, Quantum Theory of Solids. Wiley (1987). Google Scholar

  • [17] H. Kober, On fractional integrals and derivatives. Quarterly J. of Mathematics (Oxford Ser.) 11, No 1 (1940), 193–211. http://dx.doi.org/10.1093/qmath/os-11.1.193CrossrefGoogle Scholar

  • [18] A. A. Michelson and E. W. Morley, On the relative motion of the earth and the luminiferous ether American J. of Science 34 (1887), 333–345. http://dx.doi.org/10.2475/ajs.s3-34.203.333CrossrefGoogle Scholar

  • [19] A. Mielke and T. Roubicek, A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003), 571–597; doi:10.1137/S1540345903422860. http://dx.doi.org/10.1137/S1540345903422860CrossrefGoogle Scholar

  • [20] G. Pagnini, Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117–127; DOI: 10.2478/s13540-012-0008-1; http://link.springer.com/article/10.2478/s13540-012-0008-1. http://dx.doi.org/10.2478/s13540-012-0008-1CrossrefGoogle Scholar

  • [21] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367–386; http://www.math.bas.bg/~fcaa/; and Corrections to Figure 4 in: Fract. Calc. Appl. Anal. 6, No 1 (2003), 109–110. Google Scholar

  • [22] M. Riesz, L’integrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81 (1949), 1–223. http://dx.doi.org/10.1007/BF02395016CrossrefGoogle Scholar

  • [23] I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory. North-Holland Publ. Co., Amsterdam (1966). Google Scholar

  • [24] T. Young, The Bakerian lecture: On the theory of light and colours. Philosophical Transactions of the Royal Society of London 92 (1802), 12–48. http://dx.doi.org/10.1098/rstl.1802.0004CrossrefGoogle Scholar

About the article

Published Online: 2014-09-03

Published in Print: 2014-12-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-014-0221-1.

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© 2014 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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