Commuting ordinary differential operators with polynomial coefficients and automorphisms of the first Weyl algebra

Abstract

In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any \(n\ge 1\) there is a pair of self-adjoint commuting ordinary differential operators of rank two \(L_4=(\partial_x^2+V(x))^2+W(x)\), \(L_{6}\), where \(W(x),V(x)\) are polynomials of degree \(n\) and \(n+2\). We also prove that there are hyperelliptic spectral curves with the infinite spaces of orbits.