Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators

Morales, Juan J.; Rueda, Sonia L.; Zurro, Maria-Angeles

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1287-1324.

Ce travail est une étude Galoisienne du problème spectral $L\Psi =\lambda \Psi$, pour les opérateurs différentiels algébro-géométriques du second ordre $L$, avec des coefficients dans un corps différentiel, dont le corps de constantes $C$ est algébriquement clos et de caractéristique zéro. Notre approche considère le paramètre spectral $\lambda$ une variable algébrique sur $C$, ce qui amène à considérer un nouveau corps de coefficients pour $L-\lambda$, dont le corps de constantes est le champ $C\left(\Gamma \right)$ de la courbe spectrale $\Gamma$. Puisque $C\left(\Gamma \right)$ n’est plus algébriquement clos, le besoin se fait sentir d’une nouvelle structure algébrique, générée par les solutions du problème spectral sur $\Gamma$, appelée « Corps spectral de Picard–Vessiot » de $L-\lambda$. On prouve un théorème d’existence en utilisant l’algèbre différentielle, permettant de retrouver la théorie classique de Picard–Vessiot pour chaque $\lambda ={\lambda }_0$. Pour les courbes spectrales rationnelles, on établit le cadre algébrique approprié pour résoudre $L\Psi =\lambda \Psi$ de manière analytique et pour utiliser l’intégration symbolique. Nous illustrons nos résultats pour les solitons de Rosen-Morse.

This work is a Galoisian study of the spectral problem $L\Psi =\lambda \Psi$, for an algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose field of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $\lambda$ as an algebraic variable over $C$, forcing the consideration of a new field of coefficients for $L-\lambda$, whose field of constants is the field $C\left(\Gamma \right)$ of the spectral curve $\Gamma$. Since $C\left(\Gamma \right)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over $\Gamma$, called “Spectral Picard–Vessiot field” of $L-\lambda$. An existence theorem is proved using differential algebra, allowing to recover classical Picard–Vessiot theory for each $\lambda ={\lambda }_0$. For rational spectral curves, the appropriate algebraic setting is established to solve $L\Psi =\lambda \Psi$ analytically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

Classification : 12H05, 34M15
Keywords: Picard–Vessiot extension, Liouvillian extension, algebro-geometric operator, spectral curve
Mot clés : Extension de Picard–Vessiot, extension liouvillienne, opérateur algébro-géométrique, courbe spectrale

[1] Acosta-Humánez, Primitivo B.; Morales-Ruiz, Juan J.; Weil, Jacques-Arthur Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys., Volume 67 (2011) no. 3, pp. 305-374 | DOI | Zbl

[2] Arreche, Carlos E. On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters, J. Symb. Comput., Volume 75 (2016), pp. 25-55 | DOI | MR | Zbl

[3] Beardon, Alan F.; Ng, T. W. Parametrizations of algebraic curves, Ann. Acad. Sci. Fenn., Math., Volume 31 (2006) no. 2, pp. 541-554 | MR

[4] Belokolos, Eugene D.; Bobenko, Alexander I.; Enolskiĭ, Viktor Z.; Its, Alexander R.; Matveev, Vladimir B. Algebro-geometric approach in the theory of integrable equations, Springer Series in Nonlinear Dynamics, Springer, 1994

[5] Braveman, Alexander; Etingof, Pavel; Gaitsgory, Dennis Quantum integrable systems and differential Galois theory, Transform. Groups, Volume 2 (1997), pp. 31-56 | MR

[6] Brezhnev, Yurii V. On the uniformization of algebraic curves, Mosc. Math. J., Volume 8 (2008) no. 2, pp. 233-2716 | DOI | MR

[7] Brezhnev, Yurii V. Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials, Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics (Contemporary Mathematics), Volume 563, American Mathematical Society, 2012, pp. 1-31 | MR | Zbl

[8] Brezhnev, Yurii V. Elliptic solitons, Fuchsian equations, and algorithms, St. Petersbg. Math. J., Volume 24 (2013) no. 4, pp. 555-574 | DOI | MR | Zbl

[9] Bronstein, Manuel Symbolic integration I: Transcendental functions (Vol. 1), Springer Science & Business Media, Springer, 2013

[10] Burchnall, J. L.; Chaundy, Theodore W. Commutative ordinary differential operators, Proc. Lond. Math. Soc., Volume 21 (1923), pp. 420-440 | DOI | MR | Zbl

[11] Burchnall, J. L.; Chaundy, Theodore W. Commutative ordinary differential operators II. The identity $P^n=Q^m$, Proc. R. Soc. Lond., Volume 134 (1931), pp. 471-485 | Zbl

[12] Cassidy, Phyllis J.; Singer, Michael F. Galois theory of parameterized differential equations and linear differential algebraic groups, Differential equations and quantum groups (IRMA Lectures in Mathematics and Theoretical Physics), Volume 9, European Mathematical Society, 1991, pp. 113-157 | Zbl

[13] Chardin, Marc Differential resultants and subresultants, Fundamentals of computation theory (Lecture Notes in Computer Science), Volume 529, Springer, 1991, pp. 471-485 | MR | Zbl

[14] Cox, David; Little, John; O’Shea, Donal Ideals, varieties and algorithms. An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer, 2007 | Zbl

[15] Crespo, Teresa; Hajto, Zbigniew Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, 122, American Mathematical Society, 2011

[16] Dickey, Leonid A. Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, 26, World Scientific, 2003 | DOI

[17] Drach, Jules Détermination des cas de réduction de l’équation différentielle $\frac{d^{2}y}{d{x}^2}=[\varphi \left(x\right)+h]y$, C. R. Math. Acad. Sci. Paris, Volume 168 (1919) no. 1, pp. 47-50

[18] Drach, Jules Sur l’intégration par quadrature de l’équation $\frac{d^{2}y}{d{x}^2}=[\varphi \left(x\right)+h]y$, C. R. Math. Acad. Sci. Paris, Volume 168 (1919), pp. 337-340

[19] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett., Volume 19 (1967) no. 19, pp. 1095-1097 | DOI | Zbl

[20] Gel’fand, Israil’ M.; Dikii, Leonid A. Asymptotic behaviour of the resolvent of Sturm–Liouville equations and the algebra of the Korteweg–de Vries equations, Russ. Math. Surv., Volume 30 (1975) no. 5, pp. 77-113 | Zbl

[21] Gesztesy, Fritz; Holden, Helge Soliton equations and their algebro-geometric solutions: Volume 1, $(1+1)$-dimensional continuous models, Cambridge Studies in Advanced Mathematics, 79, Cambridge University Press, 2003 | DOI

[22] Goodearl, Kenneth R. Centralizers in differential, pseudo-differential and fractional differential operator rings, Rocky Mt. J. Math., Volume 13 (1983) no. 4, pp. 573-618 | Zbl

[23] Grigorenko, N. V. Algebraic-geometric operators and Galois differential theory, Ukr. Math. J., Volume 61 (2009), pp. 14-29 | DOI | MR | Zbl

[24] Halphen, Georges H. Traité des fonctions elliptiques et de leurs applications, Gauthier-Villars, 1886

[25] Hardouin, Charlotte; Minchenko, Andrei; Ovchinnikov, Alexey Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence, Math. Ann., Volume 368 (2017), pp. 587-632 | DOI | MR | Zbl

[26] Harris, Joe Algebraic geometry. A first course, Graduate Texts in Mathematics, Springer, 1992 no. 133 | DOI | Zbl

[27] Hermite, Charles Sur l’équation de Lamé, Oeuvres of Charles Hermite, III, Gauthier-Villars, 1912

[28] Hille, Einar Ordinary differential equations in the complex domain, Pure and Applied Mathematics, John Wiley & Sons, 1976

[29] Kaplansky, Irving An introduction to differential algebra, Hermann, 1976

[30] Kolchin, Ellis R. Differential algebra and algebraic groups, Pure and Applied Mathematics, Academic Press Inc., 1973 no. 54 | Zbl

[31] Krichever, Igor’ M. Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl., Volume 11 (1977), pp. 12-26 | DOI | MR | Zbl

[32] Krichever, Igor’ M. Commutative rings of ordinary linear differential operators, Funct. Anal. Appl., Volume 12 (1978) no. 3, pp. 175-185 | DOI | MR | Zbl

[33] Krichever, Igor’ M.; Novikov, Sergei P. Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2, Dokl. Akad. Nauk SSSR, Volume 247 (1979), pp. 33-37 translation in Sov. Math., Dokl. 20 (1979), p. 650-654 | Zbl

[34] Krichever, Igor’ M.; Novikov, Sergei P. Holomorphic bundles over algebraic curves and nonlinear equations, Russ. Math. Surv., Volume 35 (1980) no. 6, pp. 53-79 | DOI | MR | Zbl

[35] Li, Ziming A subresultant theory for Ore polynomials with applications, Proceedings of the 1998 international symposium on symbolic and algebraic computation, ACM Press, 1998, pp. 132-139 | MR | Zbl

[36] Łojasiewicz, Stanisław Introduction to complex analytic geometry, Birkhäuser, 1991 | DOI

[37] Matveev, Vladimir B. 30 years of finite-gap integration theory, Philos. Trans. R. Soc. Lond., Ser. A, Volume 366 (2008), pp. 837-875 | MR | Zbl

[38] McCallum, Scott; Winkler, Franz Resultants: algebraic and differential, Techn. Rep. J. Kepler University (2018), RISC18-08, 21 pages

[39] Mitschi, Claude; Singer, Michael F. Monodromy groups of parameterized linear differential equations with regular singularities, Bull. Lond. Math. Soc., Volume 44 (2011) no. 5, pp. 913-930 | DOI | MR | Zbl

[40] Morales-Ruiz, Juan J. Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, 179, Birkhäuser, 1999 | DOI

[41] Morales-Ruiz, Juan J. Picard–Vessiot theory and integrability, J. Geom. Phys., Volume 87 (2015), pp. 314-343 | DOI | MR | Zbl

[42] Morales-Ruiz, Juan J.; Rueda, Sonia L.; Zurro, Maria-Angeles Factorization of KdV Schrödinger operators using differential subresultants, Adv. Appl. Math., Volume 120 (2020), 102065, 31 pages | Zbl

[43] Motonaga, Shoya; Yagasaki, Kazuyuki Nonintegrability of parametrically forced nonlinear oscillators, Regul. Chaotic Dyn., Volume 23 (2018) no. 3, pp. 291-303 | DOI | MR | Zbl

[44] Mumford, David An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de Vries equation and related non-linear equations, Proceedings of the international symposium on algebraic geometry (Kyoto, 1977), Kinokuniya Book Store, 1978, pp. 115-153

[45] Novikov, Sergei P. The periodic problem for the Korteweg–de Vries equation, Funct. Anal. Appl., Volume 8 (1974) no. 3, pp. 236-246 | DOI | Zbl

[46] Olver, Peter J. Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer, 1986 | DOI

[47] Previato, Emma Another algebraic proof of Weil’s reciprocity, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 2 (1991) no. 2, pp. 167-171 | MR | Zbl

[48] Previato, Emma; Rueda, Sonia L.; Zurro, Maria-Angeles Commuting ordinary differential operators and the Dixmier test, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 15 (2019), 101, 23 pages | MR | Zbl

[49] van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2012

[50] Ritt, Joseph. F. Differential algebra, Colloquium Publications, 33, American Mathematical Society, 1950 | MR

[51] Schur, Issai Über vertauschbare lineare Differentialausdrücke, Sitzungsber. Berl. Math. Ges., Volume 3 (1904) no. 8, pp. 2-8 | Zbl

[52] Sendra, J. Rafael; Winkler, Franz; Pérez-Díaz, Sonia Rational algebraic curves: A computer algebra approach, Algorithms and Computation in Mathematics, Springer, 2007 no. 22 | Zbl

[53] Shafarevich, Igor R. Basic algebraic geometry, 1 and 2, Springer, 1994 | DOI | MR

[54] Ulmer, Felix; Weil, Jacques-Arthur Note on Kovacic’s algorithm, J. Symb. Comput., Volume 22 (1996) no. 2, pp. 179-200 | DOI | MR | Zbl

[55] Veselov, Alexander P. On Darboux–Treibich–Verdier potentials, Lett. Math. Phys., Volume 96 (2011) no. 1, pp. 209-216 | DOI | MR | Zbl

[56] Weikard, Rudi On commuting differential operators, Electron. J. Differ. Equ., Volume 2000 (2000), 19, 11 pages | MR

[57] Whittaker, Edmund T.; Watson, George N. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Cambridge University Press, 1996

[58] Wilson, George Algebraic curves and soliton equations, Geometry today (Progress in Mathematics), Volume 60, Birkhäuser, 1985, pp. 303-329 | MR | Zbl