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

SIGMA 13 (2017), 049, 23 pages      arXiv:1609.06247

On the Spectra of Real and Complex Lamé Operators

William A. Haese-Hill a, Martin A. Hallnäs b and Alexander P. Veselov a
a) Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
b) Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden

Received April 04, 2017, in final form June 21, 2017; Published online July 01, 2017

We study Lamé operators of the form $$ L = -\frac{d^2}{dx^2} + m(m+1)\omega^2\wp(\omega x+z_0),$$ with $m\in\mathbb{N}$ and $\omega$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $\omega$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.

Key words: Lamé operators; finite-gap operators; spectral theory; non-self-adjoint operators.

pdf (1072 kb)   tex (579 kb)


  1. Batchenko V., Gesztesy F., On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potentials, J. Anal. Math. 95 (2005), 333-387, math.SP/0312200.
  2. Belokolos E.D., Enolskii V.Z., Reduction of Abelian functions and algebraically integrable systems. II, J. Math. Sci. 108 (2002), 295-374.
  3. Birnir B., Complex Hill's equation and the complex periodic Korteweg-de Vries equations, Comm. Pure Appl. Math. 39 (1986), 1-49.
  4. Davies E.B., Pseudo-spectra, the harmonic oscillator and complex resonances, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 585-599.
  5. Davies E.B., Non-self-adjoint differential operators, Bull. London Math. Soc. 34 (2002), 513-532.
  6. Drazin P.G., Johnson R.S., Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989.
  7. Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russ. Math. Surv. 31 (1976), no. 1, 59-146.
  8. Erdélyi A., On Lamé functions, Philos. Mag. 31 (1941), 123-130.
  9. Gesztesy F., Weikard R., Floquet theory revisited, in Differential Equations and Mathematical Physics (Birmingham, AL, 1994), Int. Press, Boston, MA, 1995, 67-84.
  10. Gesztesy F., Weikard R., Picard potentials and Hill's equation on a torus, Acta Math. 176 (1996), 73-107.
  11. Grosset M.P., Veselov A.P., Elliptic Faulhaber polynomials and Lamé densities of states, Int. Math. Res. Not. 2006 (2006), 62120, 31 pages, math-ph/0508066.
  12. Grosset M.P., Veselov A.P., Lamé equation, quantum Euler top and elliptic Bernoulli polynomials, Proc. Edinb. Math. Soc. 51 (2008), 635-650, math-ph/0508068.
  13. Hermite C., Sur quelques applications des fonctions elliptique, C. R. Acad. Sci. Paris 85 (1877), 689-695, 728-732, 821-826.
  14. Ince E.L., Further investigations into the periodic Lamé functions, Proc. Roy. Soc. Edinburgh 60 (1940), 83-99.
  15. Kramers H.A., Ittmann G.P., Zur Quantelung des asymmetrischen Kreisels, Z. Phys. 53 (1929), 553-565.
  16. Kramers H.A., Ittmann G.P., Zur Quantelung des asymmetrischen Kreisels. II, Z. Phys. 58 (1929), 217-231.
  17. Lamé G., Sur les surfaces isothermes dans les corps homogènes en équilibre de température, J. Math. Pures Appl. 2 (1837), 147-188.
  18. Lax P.D., Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28 (1975), 141-188.
  19. Magnus W., Winkler S., Hill's equation, Interscience Tracts in Pure and Applied Mathematics, Vol. 20, Interscience Publishers John Wiley & Sons  New York - London - Sydney, 1966.
  20. Milnor J., Morse theory, Annals of Mathematics Studies, Vol. 51, Princeton University Press, Princeton, N.J., 1963.
  21. Novikov S.P., The periodic problem for the Korteweg-de vries equation, Funct. Anal. Appl. 8 (1974), 236-246.
  22. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, available at
  23. Pastur L.A., Tkachenko V.A., On the geometry of the spectrum of the one-dimensional Schrödinger operator with periodic complex-valued potential, Math. Notes 50 (1991), 1045-1050.
  24. Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York - London, 1978.
  25. Reed M., Simon B., Methods of modern mathematical physics. I. Functional analysis, 2nd ed., Academic Press, Inc., New York, 1980.
  26. Rofe-Beketov F.S., On the spectrum of non-selfadjoint differential operators with periodic coefficients, Soviet Math. Dokl. 4 (1963), 1563-1566.
  27. Szeg\Ho G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
  28. Takemura K., Analytic continuation of eigenvalues of the Lamé operator, J. Differential Equations 228 (2006), 1-16, math.CA/0311307.
  29. Tkachenko V.A., Spectral analysis of the one-dimensional Schrödinger operator with periodic complex-valued potential, Soviet Math. Dokl. 5 (1964), 413-415.
  30. Weikard R., On Hill's equation with a singular complex-valued potential, Proc. London Math. Soc. 76 (1998), 603-633.
  31. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  32. Zabrodin A., On the spectral curve of the difference Lamé operator, Int. Math. Res. Not. 1999 (1999), 589-614, math.QA/9812161.

Previous article  Next article   Contents of Volume 13 (2017)