Abstract
We study examples of formally self-adjoint commuting ordinary differential operators of order 4 or 4g + 2 whose coefficients are analytic on ℂ. We prove that these operators do not commute with the operators of odd order, justifying rigorously that these operators are of rank 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mironov A. E., “Self-adjoint commuting ordinary differential operators,” Invent. Math. (published on-line) DOI 10.1007/ s00222-013-0486-8.
Mironov A. E., “Periodic and rapid decay rank two self-adjoint commuting differential operators,” arXiv: 1302.5735.
Davletshina V. N., “On self-adjoint commuting differential operators of rank two,” Siberian Electron. Math. Reports, 10, 109–112 (2013).
Burchnall J. L. and Chaundy I. W., “Commutative ordinary differential operators,” Proc. London Math. Soc. Ser., 2, No. 21, 420–440 (1923).
Krichever I. M., “Integration of nonlinear equations by the methods of algebraic geometry,” Funct. Anal. Appl., 11, No. 1, 12–26 (1977).
Drinfeld V. G., “Commutative subrings of certain noncommutative rings,” Funct. Anal. Appl., 11, No. 1, 9–12 (1977).
Dixmier J., “Sur les algebres de Weyl,” Bull. Soc. Math. France, 96, 209–242 (1968).
Krichever I. M., “Commutative rings of ordinary linear differential operators,” Funct. Anal. Appl., 12, No. 3, 175–185 (1978).
Krichever I. M. and Novikov S. P., “Holomorphic bundles over algebraic curves and non-linear equations,” Russian Math. Surveys, 35, No. 6, 53–79 (1980).
Novikov S. P. and Grinevich P. G., “Spectral theory of commuting operators of rank two with periodic coefficients,” Funct. Anal. Appl., 16, No. 1, 19–20 (1982).
Grünbaum F., “Commuting pairs of linear ordinary differential operators of orders four and six,” Phys. D, 31, No. 3, 424–433 (1988).
Latham G., “Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV,” Appl. Math. Lett., 8, No. 6, 73–78 (1995).
Latham G. and Previato E., “Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations,” Acta Appl. Math., 39, 405–433 (1995).
Previato E. and Wilson G., “Differential operators and rank 2 bundles over elliptic curves,” Compositio Math., 81, No. 1, 107–119 (1992).
Mokhov O. I., “Commuting differential operators of rank 3 and nonlinear differential equations,” Math. USSR-Izv., 35, No. 3, 629–655 (1990).
Mironov A. E., “A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2,” Sb. Math., 195, No. 5, 711–722 (2004).
Mironov A. E., “On commuting differential operators of rank 2,” Siberian Electron. Math. Reports, 6, 533–536 (2009).
Mironov A. E., “Commuting rank 2 differential operators corresponding to a curve of genus 2,” Funct. Anal. Appl., 39, No. 3, 240–243 (2005).
Zuo D., “Commuting differential operators of rank 3 associated to a curve of genus 2,” SIGMA, 8, No. 044, 1–11 (2012).
Mokhov O. I., “Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients,” arXiv: 1201.5979.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Davletshina V.N. and Shamaev E.I.
The authors were supported by a Grant of the Government of the Russian Federation (Agreement No. 14.B25. 31.0029).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 4, pp. 744–749, July–August, 2014.
Rights and permissions
About this article
Cite this article
Davletshina, V.N., Shamaev, E.I. On commuting differential operators of rank 2. Sib Math J 55, 606–610 (2014). https://doi.org/10.1134/S003744661404003X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S003744661404003X