Manchester Geometry Seminar 2005/2006

If two ordinary differential operators L₁ and L₂ commute, then their common eigenfunctions are parametrized by points of the spectral curve Γ. The dimension r of the space of eigenfunctions corresponding to a general point of Γ is called the rank of the pair L₁, L₂. For operators of rank 1, the coefficients of L_i can be expressed via theta functions of the Jacobi variety of Γ. The problem of finding operators of rank r>1 is yet to be solved for the general case. Novikov and Krichever found rank 2 operators corresponding to a curve of genus 1. Mokhov used the approach of Novikov and Krichever and found operators of rank 3 and genus 1.

We give a method for constructing operators of rank 2 and genus 2. By this method one can construct operators with polynomial coefficients and formally self-adjoint.