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

Grinevich, P. G.; Novikov, S. P.

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

Keywords:

commuting ordinary differential operators; algebro-geometric operators; finite-gap potentials; Burchnall-Chaundy problem

Citations:

Zbl 0299.35017