Galoisian approach to complex oscillation theory of some Hill equations

  • Yik-Man Chiang
  • Guo-Fu Yu


We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponent of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. These are solutions obtainable by suitable differential field extension constructions. In particular, we establish a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of a biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality, and satisfy Fredholm integral equations over suitable integration regions in $\mathbb{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.


Acosta-Humánez, P. and Blázquez-Sanz, D., Non-integrability of some Hamiltonians with rational potentials, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2-3, 265–293.

Acosta-Humánez, P. B., Morales-Ruiz, J. J., and Weil, J.-A., Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys. 67 (2011), no. 3, 305–374.

Arscott, F. M., Periodic differential equations. An introduction to Mathieu, Lamé, and allied functions, International Series of Monographs in Pure and Applied Mathematics, no. 66, Macmillan Co., New York, 1964.

Arscott, F. M., Polynomial solutions of differential equations with bi-orthogonal properties, in “Conference on the Theory of Ordinary and Partial Differential Equations (Univ. Dundee, Dundee, 1972)'', Lecture Notes in Math., no. 280, Springer, Berlin, 1972, pp. 202--206.

Bank, S. B., On the explicit determination of certain solutions of periodic differential equations, Complex Variables Theory Appl. 23 (1993), no. 1-2, 101–121.

Bank, S. B. and Laine, I., On the oscillation theory of $f^\prime \prime +Af=0$ where $A$ is entire, Trans. Amer. Math. Soc. 273 (1982), no. 1, 351–363.

Bank, S. B. and Laine, I., Representations of solutions of periodic second order linear differential equations, J. Reine Angew. Math. 344 (1983), 1–21.

Bank, S. B., Laine, I., and Langley, J. K., On the frequency of zeros of solutions of second order linear differential equations, Results Math. 10 (1986), no. 1-2, 8–24.

Blázquez-Sanz, D. and Yagasaki, K., Galoisian approach for a Sturm-Liouville problem on the infinite interval, Methods Appl. Anal. 19 (2012), no. 3, 267–288.

Chiang, Y.-M., On the zero-free solutions of linear periodic differential equations in the complex plane, Results Math. 38 (2000), no. 3-4, 213–225.

Chiang, Y.-M. and Ismail, M. E. H., On value distribution theory of second order periodic ODEs, special functions and orthogonal polynomials, Canad. J. Math. 58 (2006), no. 4, 726–767, Corrigendum: Canad. J. Math. 62 (2010), no. 2, 261.

Chiang, Y.-M. and Luo, X., Complex oscillatory theory and semi-finite gap problems of Whittaker-Hill equation, forthcoming.

Dunham, J. L., The energy levels of a rotating vibrator, Phys. Rev. 41 (1932), 721–731.

Duval, A. and Loday-Richaud, M., Kovačič's algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput. 3 (1992), no. 3, 211–246.

Eastham, M. S. P., The spectral theory of periodic differential equations, Texts in Mathematics (Edinburgh), Scottish Academic Press, Edinburgh; Hafner Press, New York, 1973.

Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher transcendental functions, vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.

Fauvet, F., Ramis, J.-P., Richard-Jung, F., and Thomann, J., Stokes phenomenon for the prolate spheroidal wave equation, Appl. Numer. Math. 60 (2010), no. 12, 1309–1319.

Felder, G. and Willwacher, T., Jointly orthogonal polynomials, J. Lond. Math. Soc. (2) 91 (2015), no. 3, 750–768.

Garabedian, P. R., Partial differential equations, second ed., Chelsea Publishing Co., New York, 1986.

Hayman, W. K., Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

Heine, E., Handbuch der Kugelfunctionen, Theorie und Anwendungen, vol. 1, G. Reimer, Berlin, 1878.

Hobson, E. W., The theory of spherical and ellipsoidal harmonics, Cambridge Univ. Press, 1931, reprinted 2012.

Ince, E. L., On the connexion between linear differential systems and integral equations, Proc. R. Soc. Edinburgh 42 (1919), 43–53.

Ince, E. L., Ordinary differential equations, Longmans, Green & Co, London, 1927.

Kimura, T., On Riemann's equations which are solvable by quadratures, Funkcial. Ekvac. 12 (1969), 269–281.

Kolchin, E. R., Differential algebra and algebraic groups, Pure and Applied Mathematics, no. 54, Academic Press, New York-London, 1973.

Kovacic, J. J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), no. 1, 3–43.

Laine, I., Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993.

Magnus, W., Monodromy groups and Hill's equation, Comm. Pure Appl. Math. 29 (1976), no. 6, 691–706.

Magnus, W. and Winkler, S., Hill's equation, Dover Publications, Inc., New York, 1979.

Maroni, P., Sur quelques relations intégrales entre les solutions de l'équation biconfluente de Heun, Ann. Inst. H. Poincaré Sect. A (N.S.) 30 (1979), no. 4, 315–332.

Masson, D., The rotating harmonic oscillator eigenvalue problem. I. Continued fractions and analytic continuation, J. Math. Phys. 24 (1983), no. 8, 2074–2088.

Morales, J. J. and Simó, C., Picard-Vessiot theory and Ziglin's theorem, J. Differential Equations 107 (1994), no. 1, 140–162.

Morales-Ruiz, J. J., Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, vol. 179, Birkhäuser Verlag, Basel, 1999.

Morales-Ruiz, J. J., Picard-Vessiot theory and integrability, J. Geom. Phys. 87 (2015), 314–343.

Ronveaux, A. (ed.), Heun's differential equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

Rovder, J., Zeros of the polynomial solutions of the differential equation $xy^\prime \prime +(\beta _0+\beta _1x+\beta _2x^2)y^\prime +(\gamma -n\beta _2x)y=0$, Mat. Časopis Sloven. Akad. Vied 24 (1974), 15–20.

Shimomura, S., Oscillation results for $n$-th order linear differential equations with meromorphic periodic coefficients, Nagoya Math. J. 166 (2002), 55–82.

Turbiner, A. V., Quasi-exactly-solvable problems and $\mathrm{sl}(2)$ algebra, Comm. Math. Phys. 118 (1988), no. 3, 467–474.

Whittaker, E. T., On a class of differential equations whose solutions satisfy integral equations, Proc. Edinb. Math. Soc. 33 (1915), 14–23.

Whittaker, E. T., On Lamé's differential equation and ellipsoidal harmonic, Proc. Lond. Math. Soc. (2) 14 (1915), 260–268.

Yang, L., Value distribution theory, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1993.
How to Cite
Chiang, Y.-M., & Yu, G.-F. (2019). Galoisian approach to complex oscillation theory of some Hill equations. MATHEMATICA SCANDINAVICA, 124(1), 102-131.