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The spectrum of difference operators and algebraic curves

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Acta Mathematica

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References

  1. Abraham, R. &Marsden, J.,Foundations of Mechanics. Benjamin, San Francisco, 1978.

    MATH  Google Scholar 

  2. Adler, M., On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-De Vries type equations.Inventiones Math. (1979).

  3. Adler, M. & Van Moerbeke, P., The kac-Moody extension for the classical groups and algebraic curves. To appear.

  4. Altman, A. &Kleiman, S., Compactifying the Jacobian.Bull. A.M.S., 82 (1976), 947–949.

    Article  MathSciNet  MATH  Google Scholar 

  5. Altman, A. & Kleiman, S., Compactifying the Picard scheme.Advances in Math. To appear.

  6. Burchnall, J. L. &Chaundy, T. W., Commutative ordinary differential operators.Proc. London Math. Soc., 81 (1922), 420–440;Proc. Royal Soc. London (A), 118 (1928), 557–593 (with a note by H. F. Baker);Proc. Royal Soc. London (A), 134 (1931), 471–485.

    Google Scholar 

  7. De Souza, M., Compactifying the Jacobian. Thesis at the Tata Institute (1973).

  8. Dikii, L. A. & Gel'fand, I. M., Fractional Powers of Operators and Hamiltonian systems.Funk. Anal. Priloz., 10 (1976).

  9. Drinfeld, V. G., Elliptic modules.Mat. Sbornik, 94 (1974); translation 23 (1976), 561.

  10. Dubrovin, B. A., Matveev, V. B. & Novikov, S. P., Non-linear equations of the Kortewegde Vries type, finite zone linear operators, and Abelian manifolds.Uspehi Mat. Nauk, 31 (1976);Russian Math. Surveys 31 (1976), 55–136.

  11. Fay, J.,Theta functions on Riemann surfaces. Springer Lecture Notes 352 (1973).

  12. Kac, M. &Van Moerbeke, P., On periodic Toda lattices.Proc. Nat. Acad. Sci. U.S.A., 72 (1975), 1627–1629, and A complete solution of the periodic Toda problem.Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 2875–2880.

    MathSciNet  MATH  Google Scholar 

  13. Kempf, G., Knudsen, P., Mumford, D. &Saint-Donat, B. Toroidal embeddings I. Berlin-Heidelberg-New York: Springer vol. 339 (1973).

    MATH  Google Scholar 

  14. Kostant, B., Quantization and unitary representation.Lectures on Modern Analysis and Applications III, Berlin-Heidelberg-New York, Springer vol. 170 (1970).

    Google Scholar 

  15. Krichever, I. M., Algebro-geometric construction of the Zaharov-Shabat equations and their periodic solutions.Sov. Math. Dokl., 17 (1976), 394–397.

    MATH  Google Scholar 

  16. Krichever I. M.,Uspekhi. Mat. Nauk, (1978).

  17. —, Methods of Algebraic geometry in the theory of non-linear equations.Uspekhi Mat. Nauk, 32: 6 (1977), 183–208. Translation:Russian Math. Surveys, 32: 6 (1977), 185–213.

    MATH  Google Scholar 

  18. McKean, H. P. &Van Moerbeke, P., The spectrum of Hill's equation.Inventiones Math., 30 (1973), 217–274.

    Article  Google Scholar 

  19. —, Sur le spectre de quelques operateurs et les varietes de JacobiSem Bourbaki, 1975–76, No, 474, 1–15.

    Google Scholar 

  20. McKean, H. P. & Van Moerbeke, P. About Toda and Hill curves.Comm. Pure Appl. Math., (1979) to appear.

  21. McKean, H. P. &Trubowitz, E., The spectrum of Hill's equation, in the presence of infinitely many bands.Comm. Pure. Appl. Math., 29 (1976), 143–226.

    MathSciNet  MATH  Google Scholar 

  22. Van Moerbeke, P., The spectrum of Jacobi matrices.Inventiones Math., 37 (1976), 45–81.

    Article  MATH  Google Scholar 

  23. Van Moerbeke, P., About isospectral deformations of discrete Laplacians.Proc. of a conference on nonlinear analysis, Calgary, June 1978. Springer Lecture Notes, 1979.

  24. Mumford, D., An algebro-geometrical construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related nonlinear equationsProc. of a Conference in Algebraic Geometry, Kyoto, 1977, publ. by Japan Math. Soc.

  25. Mumford, D. Abelian varieties, Tata Institute; Oxford University Press, (1970).

  26. Mumford, D. The Spectra of Laplace-like periodic partial difference operators and algebraic surfaces, to appear.

  27. Rosenlicht, M., Generalized Jacobian varieties.Ann. of Math., 59 (1954), 505–530.

    Article  MATH  MathSciNet  Google Scholar 

  28. Serre, J. P.,Groupes algebriques et Corps de Classes, Paris, Hermann (1959).

    MATH  Google Scholar 

  29. Siegel, C. L.,Topics in complex function theory, vol. 2. New York, Wiley (1971).

    Google Scholar 

  30. Zaharov, V. E. & Shabat, A. B., A scheme for integrating the non-linear equations of math. physics by the method of the inverse scattering problem I.Funct. Anal. and its Appl. 8 (1974) (translation 1975, p. 226).

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Authors and Affiliations

  1. Brandeis University, Waltham, Mass, USA

    Pierre Van Moerbeke

  2. Cambridge, Mass, USA

    David Mumford

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  1. Pierre Van Moerbeke
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  2. David Mumford
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Additional information

Research for this paper was partially supported by NSF Grant No. MCS-75-05576 A01.

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Van Moerbeke, P., Mumford, D. The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979). https://doi.org/10.1007/BF02392090

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  • DOI: https://doi.org/10.1007/BF02392090

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