Inventiones mathematicae

, Volume 197, Issue 2, pp 417–431 | Cite as

Self-adjoint commuting ordinary differential operators

  • Andrey E. MironovEmail author


In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

Mathematics Subject Classification (2010)

37K10 37K20 



The author is grateful to I.M. Krichever, O.I. Mokhov, S.P. Novikov and V.V. Sokolov for valuable discussions and stimulating interest.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Laboratory of Geometric Methods in Mathematical PhysicsMoscow State UniversityMoscowRussia

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