Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 035, 11 pages      arXiv:1504.07063      https://doi.org/10.3842/SIGMA.2015.035
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

On a Quantization of the Classical $\theta$-Functions

Yurii V. Brezhnev
Tomsk State University, 36 Lenin Ave., Tomsk 634050, Russia

Received January 31, 2015, in final form April 17, 2015; Published online April 28, 2015

Abstract
The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find corresponding Poisson brackets. Availability of these ingredients allows us to state the problem of a canonical quantization to these equations and disclose some important problems. In a particular case the problem is completely solvable in the sense that spectrum of the Hamiltonian can be found. The spectrum is continuous, has a band structure with infinite number of lacunae, and is determined by the Mathieu equation: the Schrödinger equation with a periodic cos-type potential.

Key words: Jacobi theta-functions; dynamical systems; Poisson brackets; quantization; spectrum of Hamiltonian.

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References

  1. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
  2. Baker H.F., Abelian functions. Abel's theorem and the allied theory of theta functions,, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
  3. Bostan A., Chèze G., Cluzeau T., Weil J.-A., Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields, arXiv:1310.2778.
  4. Brezhnev Yu.V., What does integrability of finite-gap or soliton potentials mean?, Philos. Trans. R. Soc. Lond. Ser. A 366 (2008), 923-945, nlin.SI/0505003.
  5. Brezhnev Yu.V., Non-canonical extension of $\theta$-functions and modular integrability of $\vartheta$-constants, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 689-738, arXiv:1011.1643.
  6. Brezhnev Yu.V., Lyakhovich S.L., Sharapov A.A., Dynamical systems defining Jacobi's $\vartheta$-constants, J. Math. Phys. 52 (2011), 112704, 21 pages, arXiv:1012.1429.
  7. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. III, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1955.
  8. Ince E.L., Researches into the characteristic numbers of the Mathieu equation, Proc. Roy. Soc. Edinburgh 46 (1927), 20-29.
  9. Landau L.D., Lifshitz E.M., Mechanics, Pergamon Press, Oxford - London - New York - Paris, 1960.
  10. Landau L.D., Lifshitz E.M., Quantum mechanics: non-relativistic theory, Pergamon Press, London, 1981.
  11. Lawden D.F., Elliptic functions and applications, Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York, 1989.
  12. McLachlan N.W., Theory and application of Mathieu functions, Clarendon Press, Oxford, 1947.
  13. Nayfeh A.H., Introduction to perturbation techniques, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1981.
  14. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  15. Smirnov F.A., What are we quantizing in integrable field theory?, St. Petersburg Math. J. 6 (1995), 417-428, hep-th/9307097.
  16. Tannery J., Molk J., Éléments de la théorie des fonctions elliptiques, Vol. IV, Gauthier-Villars, Paris, 1902.
  17. Vanhaecke P., Algebraic integrability: a survey, Philos. Trans. R. Soc. Lond. Ser. A 366 (2008), 1203-1224.
  18. Whittaker E.T., Watson G.N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  19. Yakubovich V.A., Starzhinskij V.M., Linear differential equations with periodic coefficients, Vols. 1, 2, John Wiley & Sons, New York - Toronto, 1975.


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