## Abstract

A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of J-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such J-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves.

## Keywords

Spectral Property Explicit Formula Theta Function Algebraic Curf
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Literature cited

- 1.V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).Google Scholar
- 2.E. D. Belokolos and V. Z. Enol'skii, “The generalized Lamb Ansatz,” Teor. Mat. Fiz.,53, No. 2, 271–282 (1982).Google Scholar
- 3.A. V. Brailov, “Complete integrability of some Euler equations and applications,” Dokl. Akad. Nauk SSSR,268, No. 5, 1043–1046 (1983).Google Scholar
- 4.A. P. Veselov and S. P. Novikov, “On Poisson brackets consistent with algebraic geometry and the Korteweg-de Vries dynamics on the set of finite-zone potentials,” Dokl. Akad. Nauk SSSR,266, No. 3, 533–537 (1982).Google Scholar
- 5.I. M. Gel'fand and L. A. Dikii, “The resolvent and Hamiltonian systems,” Funkts. Anal. Prilozhen.,11, No. 2, 11–27 (1977).Google Scholar
- 6.E. B. Gledzero, F. V. Dolzhanskii, and A. M. Obukhov, Systems of Hydrodynamic Type and Their Application [in Russian], Nauka, Moscow (1981).Google Scholar
- 7.V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Heavy Body near a Fixed Point [in Russian], Gostekhizdat, Moscow (1953).Google Scholar
- 8.P. Griffiths and J. Harris, Principles of Algebraic Geometry [Russian translation], Vol. 1, Mir, Moscow (1982).Google Scholar
- 9.L. A. Dikii, “Remarks on Hamiltonian systems connected with the rotation group,” Funkts. Anal. Prilozhen.,6, No. 4, 83–84 (1972).Google Scholar
- 10.B. A. Dubrovin, “Analytic properties of spectral data for non-self-adjoint linear operators connected with real periodic solutions of the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,265, No. 4, 789–793 (1982).Google Scholar
- 11.B. A. Dubrovin, “Completely integrable Hamiltonian systems connected with matrix operators and Abelian manifolds,” Funkts. Anal. Prilozhen.,11, No. 4, 28–41 (1977).Google Scholar
- 12.B. A. Dubrovin, “The periodic problem for the Korteweg-de Vries equation in the class of finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 3, 41–51 (1975).Google Scholar
- 13.B. A. Dubrovin, “Theta functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11–80 (1981).Google Scholar
- 14.B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR,229 No. 1, 15–18(1976).Google Scholar
- 15.B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55–136 (1976).Google Scholar
- 16.B. A. Dubrovin and S. M. Natanzon, “Real two-zone solutions of the sine-Gordon equation,” Funkts. Anal. Prilozhen.,16, No. 1, 27–43 (1982).Google Scholar
- 17.B. A. Dubrovin and S. P. Novikov, “Algebrogeometric Poisson brackets for real finitezone solutions of the sine-Gordon equation and the nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR,267, No. 6, 1295–1300 (1982).Google Scholar
- 18.B. A. Durbovin and S. P. Novikov, “Periodic and conditionally periodic analogues of multisoliton solutions of the Korteweg-de Vries equation,” Zh. Eksp. Teor. Fiz.,67, No. 12, 2131–2143 (1974).Google Scholar
- 19.B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and Applications [in Russian], Nauka, Moscow (1979).Google Scholar
- 20.V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, “A complete description of solutions of the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,219, No. 6, 1334–1337 (1974).Google Scholar
- 21.A. R. Its, “inversion of hyperelliptic integrals and integration of nonlinear differential equations,” Vestn. Leningr. Univ., No. 7, 39–46 (1976).Google Scholar
- 22.A. R. Its and V. P. Kotlyarov, “Explicit formulas for solutions of the nonlinear Schrödinger equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 965–968 (1976).Google Scholar
- 23.A. R. Its and V. B. Matveev, “Schrodinger operators with finite-zone spectrum and N-soliton solutions of the Korteweg-de Vries equation,” Teor. Mat. Fiz.,23, No. 1, 51–68 (1975).Google Scholar
- 24.V. A. Kozel and V. P. Kotlyarov, “Almost-periodic solutions of the equation u
_{tt}−u_{xx}+sin u =0,” Dokl. Akad. Nauk Ukr. SSR, A, No. 10, 878–881 (1976).Google Scholar - 25.V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk,38, No. 1, 3–67 (1983).Google Scholar
- 26.I. M. Krichever, “Algebraic curves and commuting matrix differential operators,” Funkts. Anal. Prilozhen.,10, No. 2, 75–76 (1976).Google Scholar
- 27.I. M. Krichever, “Algebrogeometric construction of the Zakharov-Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976).Google Scholar
- 28.I. M. Krichever, “Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 183–208 (1977).Google Scholar
- 29.I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations,” Usp. Mat. Nauk,35, No. 6, 47–68 (1980).Google Scholar
- 30.S. V. Manakov, “A remark on the integration of the Euler equations of the dynamics of an n-dimensional solid body,” Funkts. Anal. Prilozhen.,10, No. 4, 93–94 (1976).Google Scholar
- 31.V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).Google Scholar
- 32.A. S. Mishchenko, “Integrals of geodesic flows on Lie groups,” Funkts. Anal. Prilozhen.,4, No. 3, 73–77 (1970).Google Scholar
- 33.A. S. Mishchenko and A. T. Fomenko, “The Euler equations on finite-dimensional Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,42, 396–415 (1978).Google Scholar
- 34.S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation,” Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974).Google Scholar
- 35.S. P. Novikov (ed.), Theory of Solitons. The Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).Google Scholar
- 36.V. A. Rokhlin, “Complex topological characteristics of real algebraic curves,” Usp. Mat. Nauk,33, No. 5, 77–89 (1978).Google Scholar
- 37.V. V. Trofimov, “The Euler equations on Borel subalgebras of semisimple Lie algebras,” Izv. Akad. Nauk SSSR, Ser. Mat.,43, No. 3, 714–732 (1979).Google Scholar
- 38.I. V. Cherednik, “Algebraic aspects of two-dimensional chiral fields. II,” Itogi Nauki i Tekh. VINITI. Algebra. Topologiya. Geometriya,18, 1981, pp. 73–150.Google Scholar
- 39.I. V. Cherednik, “Differential equations for the Baker-Akhiezer functions of algebraic curves,” Funkts. Anal. Prilozhen.,12, No. 3, 45–54 (1978).Google Scholar
- 40.I. V. Cherednik, “On realness conditions in ‘finite-zone integration,’” Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).Google Scholar
- 41.I. V. Cherednik, “On regularity of ‘finite-zone’ solutions of integrable matrix differential equations,” Dokl. Akad. Nauk SSSR,266, No. 3, 593–597 (1982).Google Scholar
- 42.M. Adler and P. van Moerbeke, “Completely integrable systems, Euclidean Lie algebras, and curves,” Adv. Math.,38, No. 3, 267–317 (1980).CrossRefGoogle Scholar
- 43.M. Adler and P. van Moerbeke, “Linearization of Hamiltonian systems, Jacobi varieties, and representation theory,” Adv. Math.,38, No. 3, 318–379 (1980).CrossRefGoogle Scholar
- 44.M. Adler and P. van Moerbeke, “The algebraic integrability of a geodesic flow on S
*O*(4),” Invent. Math.,67, No. 2, 297–331 (1982).CrossRefGoogle Scholar - 45.B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Topological and algebraic geometrical methods in modern mathematical physics. II,” in: Soviet Scientific Reviews, Vol. 3, Harwood Acad. Publisher, New York (1982).Google Scholar
- 46.J. Fay, “Theta-functions on Riemann surfaces,” Lect. Notes Math., Springer,352 (1973).Google Scholar
- 47.G. Forest and D. W. McLaughlin, “Spectral theory for the periodic sine-Gordon equation: a concrete viewpoint,” J. Math. Phys.,23, No. 7, 1248 (1982).CrossRefGoogle Scholar
- 48.P. D. Lax, “Periodic solutions of the Korgeweg-de Vries equation,” Commun. Pure Appl. Math.,28, 141–188 (1975).Google Scholar
- 49.P. D. Lax, “Periodic solutions of the KdV equation,” Lect. Appl. Math.,15, 85–96 (1974).Google Scholar
- 50.V. B. Matveev, “Abelian functions and solitons,” Preprint No. 373, Inst. Teor. Phys., Wroclaw (1976).Google Scholar
- 51.H. P. McKean, “The sine-Gordon and sinh-Gordon equations on the circle,” Commun. Pure Appl. Math.,34, No. 2, 197–257 (1981).Google Scholar
- 52.H. P. McKean and P. van Moerbeke, “The spectrum of Hill's equation,” Invent. Math.,30, No. 3, 217–274 (1975).CrossRefGoogle Scholar

## Copyright information

© Plenum Publishing Corporation 1985