Common Eigenfunctions of Commuting Differential Operators of Rank $2$

Commuting differential operators of rank $2$ are considered. With each pair of commuting operators a complex curve called the spectral curve is associated. The genus of this curve is called the genus of the commuting pair. The dimension of the space of common eigenfunctions is called the rank of the commuting operators. The case of rank $1$ was studied by I. M. Krichever; there exist explicit expressions for the coefficients of commuting operators in terms of Riemann theta-functions. The case of rank $2$ and genus $1$ was considered and studied by S. P. Novikov and I. M. Krichever. All commuting operators of rank $3$ and genus $1$ were found by O. I. Mokhov. A. E. Mironov invented an effective method for constructing operators of rank $2$ and genus greater than $1$; by using this method, many diverse examples were constructed. Of special interest are commuting operators with polynomial coefficients, which are closely related to the Jacobian problem and many other problems. Common eigenfunctions of commuting operators with polynomial coefficients and smooth spectral curve are found explicitly in the present paper. This has not been done so far.