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The spectrum of difference operators and algebraic curves. (English) Zbl 0502.58032

37A30 Ergodic theorems, spectral theory, Markov operators
14H40 Jacobians, Prym varieties
47B39 Linear difference operators
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58C10 Holomorphic maps on manifolds
14K25 Theta functions and abelian varieties
32G20 Period matrices, variation of Hodge structure; degenerations
14C40 Riemann-Roch theorems
14H52 Elliptic curves
12H10 Difference algebra
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
14H15 Families, moduli of curves (analytic)
14C20 Divisors, linear systems, invertible sheaves
Full Text: DOI
[1] Abraham, R. &Marsden, J.,Foundations of Mechanics. Benjamin, San Francisco, 1978. · Zbl 0393.70001
[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). · Zbl 0393.35058
[3] Adler, M. & Van Moerbeke, P., The kac-Moody extension for the classical groups and algebraic curves. To appear. · Zbl 0491.58017
[4] Altman, A. &Kleiman, S., Compactifying the Jacobian.Bull. A.M.S., 82 (1976), 947–949. · Zbl 0336.14008 · doi:10.1090/S0002-9904-1976-14229-2
[5] Altman, A. & Kleiman, S., Compactifying the Picard scheme.Advances in Math. To appear. · Zbl 0427.14015
[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. · JFM 49.0311.03
[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. · Zbl 0326.35011
[11] Fay, J.,Theta functions on Riemann surfaces. Springer Lecture Notes 352 (1973). · Zbl 0281.30013
[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. · Zbl 0343.34003 · doi:10.1073/pnas.72.4.1627
[13] Kempf, G., Knudsen, P., Mumford, D. &Saint-Donat, B. Toroidal embeddings I. Berlin-Heidelberg-New York: Springer vol. 339 (1973). · Zbl 0271.14017
[14] Kostant, B., Quantization and unitary representation.Lectures on Modern Analysis and Applications III, Berlin-Heidelberg-New York, Springer vol. 170 (1970). · Zbl 0223.53028
[15] Krichever, I. M., Algebro-geometric construction of the Zaharov-Shabat equations and their periodic solutions.Sov. Math. Dokl., 17 (1976), 394–397. · Zbl 0361.35007
[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. · Zbl 0372.35002
[18] McKean, H. P. &Van Moerbeke, P., The spectrum of Hill’s equation.Inventiones Math., 30 (1973), 217–274. · Zbl 0319.34024 · doi:10.1007/BF01425567
[19] –, Sur le spectre de quelques operateurs et les varietes de JacobiSem Bourbaki, 1975–76, No, 474, 1–15.
[20] McKean, H. P. & Van Moerbeke, P. About Toda and Hill curves.Comm. Pure Appl. Math., (1979) to appear. · Zbl 0422.14017
[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. · Zbl 0339.34024 · doi:10.1002/cpa.3160290203
[22] Van Moerbeke, P., The spectrum of Jacobi matrices.Inventiones Math., 37 (1976), 45–81. · Zbl 0361.15010 · doi:10.1007/BF01418827
[23] Van Moerbeke, P., About isospectral deformations of discrete Laplacians.Proc. of a conference on nonlinear analysis, Calgary, June 1978. Springer Lecture Notes, 1979. · Zbl 0464.58019
[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. · Zbl 0423.14007
[25] Mumford, D. Abelian varieties, Tata Institute; Oxford University Press, (1970). · Zbl 0223.14022
[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. · Zbl 0058.37002 · doi:10.2307/1969715
[28] Serre, J. P.,Groupes algebriques et Corps de Classes, Paris, Hermann (1959). · Zbl 0097.35604
[29] Siegel, C. L.,Topics in complex function theory, vol. 2. New York, Wiley (1971). · Zbl 0211.10501
[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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