On commuting ordinary differential operators with polynomial
  coefficients corresponding to spectral curves of genus two

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Abstract

The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

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Commutative rings of ordinary linear differential operators

I. M. Krichever (1978)

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

Andrey Mironov, Alexander Zheglov (2015)

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.