Discretization of Commuting Ordinary Differential Operators of Rank 2 in the Case of Elliptic Spectral Curves

Gulnara S. Mauleshova ; Andrey E. Mironov

Informatics and Automation, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 217-229.

Résumé

We study one-point commuting difference operators of rank $2$ and establish a relationship between these operators and commuting differential operators of rank $2$ in the case of elliptic spectral curves.

Détail

Keywords: commuting difference operators, commuting differential operators.

Bibliographie

[1] V. N. Davletshina and E. I. Shamaev, “On commuting differential operators of rank 2”, Sib. Math. J., 55:4 (2014), 606–610 | DOI | MR | Zbl

[2] Dehornoy P., “Opérateurs différentiels et courbes elliptiques”, Compos. math., 43:1 (1981), 71–99 | MR | Zbl

[3] Dixmier J., “Sur les algèbres de Weyl”, Bull. Soc. math. France, 96 (1968), 209–242 | DOI | MR | Zbl

[4] V. G. Drinfel'd, “Commutative subrings of certain noncommutative rings”, Funct. Anal. Appl., 11:1 (1977), 9–12 | DOI | MR | Zbl | Zbl

[5] P. G. Grinevich, “Rational solutions for the equation of commutation of differential operators”, Funct. Anal. Appl., 16:1 (1982), 15–19 | DOI | MR | MR | Zbl

[6] Grünbaum F.A., “Commuting pairs of linear ordinary differential operators of orders four and six”, Physica D, 31:3 (1988), 424–433 | DOI | MR | Zbl

[7] A. V. Ilina and I. M. Krichever, “Triangular reductions of the $2D$ Toda hierarchy”, Funct. Anal. Appl., 51:1 (2017), 48–65 | DOI | MR | Zbl

[8] I. M. Krichever, “Algebraic curves and non-linear difference equations”, Russ. Math. Surv., 33:4 (1978), 255–256 | DOI | MR | Zbl | Zbl

[9] I. M. Krichever, “Commuting difference operators and the combinatorial Gale transform”, Funct. Anal. Appl., 49:3 (2015), 175–188 | DOI | MR | Zbl

[10] I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations”, Russ. Math. Surv., 35:6 (1980), 53–79 | DOI | MR | Zbl | Zbl

[11] I. M. Krichever and S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Russ. Math. Surv., 58:3 (2003), 473–510 | DOI | MR | Zbl

[12] Latham G.A., “Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV”, Appl. Math. Lett., 8:6 (1995), 73–78 | DOI | MR | Zbl

[13] Latham G.A., Previato E., “Darboux transformations for higher-rank Kadomtsev–Petviashvili and Krichever–Novikov equations”, Acta appl. math., 39:1–3 (1995), 405–433 | DOI | MR | Zbl

[14] G. S. Mauleshova and A. E. Mironov, “Difference Krichever–Novikov operators of rank 2”, Proc. Steklov Inst. Math., 305 (2019), 195–208 | DOI | MR | Zbl

[15] A. E. Mironov, “Discrete analogues of Dixmier operators”, Sb. Math., 198:10 (2007), 1433–1442 | DOI | MR | Zbl

[16] Mironov A.E., “Self-adjoint commuting ordinary differential operators”, Invent. math., 197:2 (2014), 417–431 | DOI | MR | Zbl

[17] A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russ. Math. Surv., 71:4 (2016), 751–779 | DOI | MR | Zbl

[18] Mironov A.E., Mauleshova G.S., “On rank two algebro-geometric solutions of an integrable chain”, Geometric methods in physics XXXVI: Proc. Workshop (Białowieża, 2017), Trends Math., Birkhäuser, Cham, 2019, 189–195 | DOI | MR | Zbl

[19] Van Moerbeke P., Mumford D., “The spectrum of difference operators and algebraic curves”, Acta math., 143 (1979), 93–154 | DOI | MR | Zbl

[20] O. I. Mokhov, “Commuting differential operators of rank 3, and nonlinear differential equations”, Math. USSR, Izv., 35:3 (1990), 629–655 | DOI | MR | Zbl

[21] O. I. Mokhov, “On commutative subalgebras of the Weyl algebra related to commuting operators of arbitrary rank and genus”, Math. Notes, 94:2 (2013), 298–300 | DOI | MR | Zbl

[22] Mokhov O.I., “Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients”, Topology, geometry, integrable systems and mathematical physics. Novikov's seminar 2012–2014, AMS Transl. Ser. 2, 234, Amer. Math. Soc., Providence, RI, 2014, 323–336 | MR | Zbl

[23] Mumford D., “An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg–de Vries equation and related non-linear equations”, Proc. Int. Symp. on Algebraic Geometry (Kyoto, 1977), Kinokuniya, Tokyo, 1978, 115–153 | MR

[24] S. P. Novikov, “Commuting operators of rank $l>1$ with periodic coefficients”, Sov. Math., Dokl., 25 (1982), 535–538 | MR | Zbl

[25] S. P. Novikov and P. G. Grinevich, “Spectral theory of commuting operators of rank two with periodic coefficients”, Funct. Anal. Appl., 16:1 (1982), 19–20 | DOI | MR | Zbl

[26] V. S. Oganesyan, “Commuting differential operators of rank 2 and arbitrary genus $g$ with polynomial coefficients”, Russ. Math. Surv., 70:1 (2015), 165–167 | DOI | MR | Zbl

[27] Pogorelov D.A., Zheglov A.B., “An algorithm for construction of commuting ordinary differential operators by geometric data”, Lobachevskii J. Math., 38:6 (2017), 1075–1092 | DOI | MR | Zbl

[28] Previato E., Rueda S.L., Zurro M.-A., “Commuting ordinary differential operators and the Dixmier test”, SIGMA. Symmetry Integrability Geom. Methods Appl., 15 (2019), 101 | MR | Zbl

[29] Previato E., Wilson G., “Differential operators and rank 2 bundles over elliptic curves”, Compos. math., 81:1 (1992), 107–119 | MR | Zbl

[30] Yamilov R., “Symmetries as integrability criteria for differential difference equations”, J. Phys. A: Math. Gen., 39:45 (2006), R541–R623 | DOI | MR | Zbl

[31] Zuo D., “Commuting differential operators of rank 3 associated to a curve of genus 2”, SIGMA. Symmetry Integrability Geom. Methods Appl., 8 (2012), 044 | MR | Zbl