##
**On \(s\)-meromorphic ordinary differential operators.**
*(English)*
Zbl 1418.34159

Russ. Math. Surv. 71, No. 6, 1143-1145 (2016); translation from Usp. Mat. Nauk 71, No. 6, 161-162 (2016).

From the text: The theory of commuting ordinary differential operators that was developed in
the 1920s is connected with the KdV equation, but this connection was not known until 1974. The KdV theory was based on the quantum mechanical scattering problem, and
solitons were constructed as reflectionless potentials. It was not known until 1974 that
they determine algebro-geometric operators (which admit ordinary differential operators
commuting with them). This was discovered in 1974 (see [S. P. Novikov, Funct. Anal. Appl. 8, 236–246 (1974; Zbl 0299.35017); translation from Funkts. Anal. Prilozh. 8, No. 3, 54–66 (1974)]) by Novikov and Lax in
solving the periodic problem in which finite-gap potentials appear as periodic analogues
of reflectionless potentials.
The soliton theory also made it possible to solve the classical Burchnall-Chaundy
problem of classifying commuting pairs of higher rank \(k > 1\).
In this paper we work only with algebro-geometric operators of rank 1.

### MSC:

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

34L05 | General spectral theory of ordinary differential operators |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |