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Information theoretic spreading measures of orthogonal functions

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Abstract

We calculate information theoretic spreading measures of orthogonal functions associated with solutions of quantum mechanical isospectral potentials. In particular, Shannon, Renyi and Fisher lengths have been evaluated for potentials isospectral to the linear harmonic oscillator and the symmetric Rosen-Morse potentials. We have also compared the behaviour of different lengths for the orthogonal functions and the associated orthogonal polynomials.

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References

  1. D.L. Pursey, New families of isospectral Hamiltonians. Phys. Rev. D 33, 1048–1055 (1986)

    Google Scholar 

  2. G. Dunne, J. Feinberg, Self-isospectral periodic potentials and supersymmetric quantum mechanics. Phys. Rev. A 57, 1271–1276 (1998)

    Google Scholar 

  3. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon, NY, 1965)

    Google Scholar 

  4. S. Flugge, Practical Quantum Mechanics (Springer, New York, 1974)

    Google Scholar 

  5. F. Cooper, A. Khare, U.P. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001)

    Book  Google Scholar 

  6. B. Bagchi, Supersymmetry in Quantum and Classical Mechanics (Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 116, 2001)

  7. G. Junker, Supersymmetric Methods in Quantum and Statistical Physics (Springer, Berlin, 1996)

    Book  Google Scholar 

  8. B. Bagchi, S.N. Biswas, A. Khare, P.K. Roy, Pramana J. Phys. 49, 199 (1997)

    Article  CAS  Google Scholar 

  9. G. Junker, P. Roy, Conditionally exactly solvable potentials: A supersymmetric construction method, arXiv:quant-ph/9803024v1 11 Mar (1998)

  10. B. Bagchi, C. Quesne, Conditionally exactly solvable potential and dual transformation in quantum mechanics, arXiv:math-ph/0401041v1 23 Jan (2004)

  11. T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978)

    Google Scholar 

  12. E.A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials. Math. USSR Sb. 32, 199–213 (1977)

    Article  Google Scholar 

  13. M.X. He a, P.E. Ricci, Information entropy of orthogonal polynomials. Appl. Math. Comput. 128, 261–274 (2002)

    Article  Google Scholar 

  14. P’oschl, E. Teller, Z. Phys. 83, 143–151 (1933)

  15. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)

    Book  Google Scholar 

  16. R.A. Fisher, Theory of statistical estimation. Proc. Camb. Philos. Soc. 22, 700–725, reprinted in Collected Papers of R.A. Fisher, edited by J.H. Bennet (University of Adelaide Press, South Australia), 1972, 15–40 (1925)

  17. A. Renyi, Probability Theory (North Holland, Amsterdam, 1970)

    Google Scholar 

  18. C.E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)

    Google Scholar 

  19. P. Sanchez-Moreno, J.S. Dehesa, D. Manzano, R.J. Yanez, Spreading lengths of Hermite polynomials. J. Comput. Appl. Math. 133, 23–46 (2001)

    Article  Google Scholar 

  20. P. Sanchez-Moreno, D. Manzano, J.S. Dehesa, Direct spreading measures of Lagurre polynomials. J. Comput. Appl. Math. 235, 1129–1140 (2011)

    Google Scholar 

  21. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, USA, 2000)

    Google Scholar 

  22. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (Dover, New York, 1970)

    Google Scholar 

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  1. Department of Mathematics, Vivekananda College, 269 D.H. Road, Kolkata, 700063, India

    D. Nath

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  1. D. Nath
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Nath, D. Information theoretic spreading measures of orthogonal functions. J Math Chem 51, 1446–1461 (2013). https://doi.org/10.1007/s10910-013-0157-4

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