1107.3356 (Andrey E. Mironov)

Self-adjoint commuting differential operators and commutative
subalgebras of the Weyl algebra    [PDF]

Andrey E. Mironov

In this paper we study self-adjoint commuting ordinary differential operators. 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 coefficients of the self-adjoint operator of order four 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 arbitrary genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

View original: http://arxiv.org/abs/1107.3356