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

TL;DR

This paper investigates rank two commuting differential operators with polynomial coefficients, exploring their automorphisms and the structure of their orbit spaces, revealing infinite orbit spaces for genus one spectral curves and constructing specific self-adjoint operators.

Contribution

It demonstrates the infinite nature of orbit spaces for fixed genus one spectral curves and constructs explicit self-adjoint commuting operators with polynomial coefficients.

Findings

Infinite orbit spaces for genus one spectral curves.

Existence of self-adjoint commuting operators with polynomial coefficients.

Construction of operators with polynomial potentials of specified degrees.

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.