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

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.

💡 Research Summary

The paper investigates self‑adjoint commuting ordinary differential operators and their connection to commutative subalgebras of the first Weyl algebra (A_{1}). The authors focus on a pair of operators (L_{4}) (order 4) and (L_{4g+2}) (order (4g+2)) that commute and have a common hyperelliptic spectral curve (\Gamma) of genus (g).

First, they establish a necessary and sufficient condition for (L_{4}) to be self‑adjoint. Writing (L_{4}) in the form
\