Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

Yıl 2019, Cilt: 2 Sayı: 1, 63 - 90, 17.06.2019

Ahmed Lesfari

https://doi.org/10.33401/fujma.540070

Cited By: 1

Öz

The aim of this paper is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and linearization of nonlinear dynamical systems. We explain how results from isospectral deformations, cohomology groups and algebraic geometry can be used to obtain insight into integrable systems. Another part will be dedicated to the study of infinite continued fractions and isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Also, the notion of algebraically completely integrable systems is explained and techniques to solve such systems are presented. Several nonlinear problems in mathematical physics illustrate these results.

Anahtar Kelimeler

difference operators, Jacobian varieties, Lax equations, Cohomology group, Continued fraction, Deformation, Integrable systems, Periodic Jacobi matrices, Spectral curves

Kaynakça

• [1] M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math., 50(3) (1979), 219-248.
• [2] M. Adler, P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras, and Curves, Adv. in Math., 38 (1980), 267-379.
• [3] M. Adler, P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math., 38 (1980), 318-379.
• [4] M. Adler, P. van Moerbeke, Algebraic completely integrable systems : a systematic approach, I, II, III, S´eminaire de Math´ematique, Rapport No 110, p.1-145, SC/MAPA - Institut de math´ematique pure et appliqu´ee, UCL, 1985.
• [5] M. Adler, P. van Moerbeke, The complex geometry of the Kowalewski-Painlev´e analysis, Invent. Math., 97 (1989), 3-51.
• [6] M. Adler, P. van Moerbeke, The Toda lattice, Dynkin diagrams, singularities and abelian varieties, Invent. Math., 103(2) (1991), 223-278.
• [7] M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic integrability, Painlev´e geometry and Lie algebras, A series of modern surveys in mathematics, Volume 47, Springer-Verlag, 2004.
• [8] M. Adler, P. van Moerbeke, The AKS theorem, A.C.I. systems and random matrix theory, Journal of Physics A: Mathematical and Theoretical, Volume 51, Number 42 (2018).
• [9] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, I. Grundlehren der mathematischenWissenschaften, 267, Springer-Verlag, New York, 1985.
• [10] V.I.: Arnold, Mathematical methods in classical mechanics, Springer-Verlag, Berlin-Heidelberg- New York, 1978, 2nd edn., Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1989.
• [11] P. Deift F. Lund, E. Trubowitz, Nonlinear Wave Equations and Constrained Harmonic Motion, Comm. Math. Phys., 74 (1980), 141-188.
• [12] B. A. Dubrovin, S. P. Novikov, Periodic and conditionally periodic analogues of multi-soliton solutions of the Korteweg-de Vries equation, Soviet Physics JETP, 40 (1974), 1058-1063.
• [13] H. Flaschka, The Toda lattice I, Existence of integrals, Phys. Rev., B 3, 9 (1974), 1924-1925 (1974).
• [14] J.-P. Franc¸oise, Integrability of quasi-homogeneous vector fields, Unpublished preprint.
• [15] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. and Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.
• [16] B. Grammaticos, B. Dorozzi, A. Ramani, Integrability of Hamiltonians with third and fourth-degree polynomial potentials, J. Math. Phys., 24 (1983), 2289-2295.
• [17] P. A. Griffiths, J. Harris, Principles of algebraic geometry, Wiley-Interscience, New-York, 1978.
• [18] P. A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations, Amer. J. Math., 107 (1985), 1445-1483.
• [19] L. Haine, Geodesic flow on SO(4) and Abelian surfaces, Math. Ann., 263 (1983), 435-472.
• [20] M. H´enon, C. Heiles, The applicability of the third integral of motion; some numerical experiments, Astron. J., 69 (1964), 73-79.
• [21] E. Hille, Ordinary differential equations in the complex domain, Wiley-Interscience, New-York, 1976.
• [22] C. G. J. Jacobi, Vorlesungen ¨uber Dynamik, K¨onigsberg lectures of 1842-1843, (reprinted by Chelsea Publishing Co., New York, 1969.
• [23] M. Kac, P. van Moerbeke, On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. in Math., 16 (1975), 160-169.
• [24] B. B. Kadomtsev, V. I. Petviashvili, On the Stability of Solitary Waves in Weakly Dispersing Media, Sov. Phys. Dokl., 15 (6) (1970), 539-541.
• [25] Y. Kato, On the spectral density of periodic Jacobi matrices, Proceedings of RIMS Symposium on Non-Linear Integrable Systems-Classical Theory and Quantum Theory, Kyoto Japan, 1981, 153-181, World Science Publishing Co., 1983.
• [26] Y. Kato, Mixed periodic Jacobi continued fractions, Nagoya Math. J., 104 (1986), 129-148.
• [27] H. Knorrer, Geodesics on the ellipso¨ıd, Invent. Math., 59 (1980), 119-143.
• [28] H. Knorrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69-78.
• [29] H. Knorrer, Integrable Hamiltonsche Systeme und algebraische Geometrie, Jahresber. Deutsch. Math.- Verein., 88 (2) (1986), 82-103.
• [30] D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.
• [31] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. Math., 34 (3) (1979), 195-338.
• [32] S. Kowalewski, Sur le probl`eme de la rotation d’un corps solide autour d’un point fixe, Acta Math., 12 (1889), 177-232.
• [33] I. M. Krichever, Algebraic-geometric construction of Zakhorov-Shabat equations and their periodic solutions, Sov. Math., Dokl., 17 (1976), 394-397 (1976).
• [34] J. L. Lagrange, M´ecanique analytique, Oeuvres de Lagrange, Serret J.A. [Darboux G.] eds., t. 11, Gauthier-Villars, Paris, 1888.
• [35] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.
• [36] A. Lesfari, Abelian surfaces and Kowalewski’s top, Ann. Scient. ´ Ecole Norm. Sup., Paris, 4e s´erie, .21 (1988), 193-223.
• [37] A. Lesfari, Abelian varieties, surfaces of general type and integrable systems, Beitr¨age Algebra Geom., 48 (1) (2007), 95-114.
• [38] A. Lesfari, Integrable systems and complex geometry, Lobachevskii J. Math., 30(4) (2009), 292-326.
• [39] A. Lesfari, Th´eorie spectrale et probl`emes non-lin´eaires, Surv. Math. Appl., 5 (2010), 151-190.
• [40] A. Lesfari, Algebraic integrability : the Adler-van Moerbeke approach, Regul. Chaotic Dyn., 16 (3-4) (2011), 187-209.
• [41] A. Lesfari, The H´enon-Heiles system as part of an integrable system, J. Adv. Res. Dyn. Control Syst., 6 (3) (2014), 24-31.
• [42] A. Lesfari, Introduction `a la g´eom´etrie alg´ebrique complexe, ´ Editions Hermann, Paris, 2015.
• [43] S. V. Manakov, Remarks on the Integrals of the Euler Equations of the n-Dimensional Heavy Top, Fund. Anal. Appl., 10 (4) (1976), 93-94.
• [44] H. P. Mc Kean, P. van Moerbeke, The Spectrum of Hill’s Equation, Invent. Math., 30 (1975), 217-274.
• [45] J. Moser, Various Aspects of Integrable Hamiltonian Systems, In: Guckenheimer J., Moser J., and Newhouse S. E., Dynamical Systems, O.I.M.E.
• Lectures, Bressanone, Italy, June 1978, Progr. Math. 8, Birkh¨auser, Boston, 233-289, 1980.
• [46] J. Moser, Geometry of quadrics and spectral theory, Lecture delivred at the symposium in honor of S.S. Chern, Berkeley, 1979. Springer, Berlin, Heidelberg, New-York, 1980.
• [47] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de-Vries equation and related non-linear equations, Intl. Symp. on Algebraic Geometry, Kyoto, (1977), 115-153.
• [48] D. Mumford, Tata lectures on theta I, II, Progress in Math., Birkha¨user, Boston, 1983.
• [49] C. Neumann, De problemate quodam mechanics, quod ad primam integralium ultraellipticorum classem revocatur, J. Reine Angew. Math., 56 (1859), 46-63.
• [50] P. Painlev´e, Oeuvres, tomes 1,2,3. ´ Edition du C.N.R.S., 1975.
• [51] H. Poincar´e, Lec¸ons de m´ecanique c´eleste, 3 tomes. Gauthier-Villars, 1905-1910, r´e´edit´e par Jacques Gabay, Paris 2003.
• [52] A. Ramani, B. Dorozzi, B. Grammaticos, B. Painlev´e conjecture revisited, Phys. Rev. Lett., 49 (1982), 1539-1541.
• [53] T. Ratiu, Euler-Poisson Equations on Lie Algebras and the N-dimensional Heavy Rigid Body, Amer. J. Math., 104 (1982), 409-448.
• [54] T. Ratiu, P. van Moerbeke, The Lagrange Rigid Body Motion, Ann. Inst. Fourier, 32(1) (1982), 211-234.
• [55] W. Symes, Systems of Toda type, inverse spectral problems and representation theory, Invent. Math., 59 (1980), 13-53.
• [56] G. Teschil, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs Volume 72, 2000.
• [57] M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. of Japan, 23 (1967), 501-506.
• [58] P. Vanhaecke, Integrable systems in the realm of algebraic geometry, Lecture Notes in Math., 1638, Springer-Verlag, Berlin, 2001.
• [59] P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81.
• [60] P. van Moerbeke, D. Mumford, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.
• [61] V. E. Zaharov, A. B. Shabat, A Scheme for Integrating the Nonlinear Equations of Math. Physics by the Method of the Inverse Scattering Problem I, Funct. Analysis and its Appl., 8 (1974), translation, 226 (1975).