Commuting Differential Operators of Rank 2 with Polynomial Coefficients

Commuting Ordinary Differential Operators and the Dixmier Test

“…1. U (x) = 0 and W (x) = 4x 2 + w 0 or W (x) = 8x 2 + w 0 , which are self-adjoint examples given in [34], with g = 1 and g = 2 respectively.…”

“…By [34,Theorem 2], this operator commutes with an operator of order 4q +2 with q ≥ g = 2. We fixed 4q+2 = 18, that is q = 4, and computed an operator M of order 18 in the centralizer C(L 4 We run the algorithm BC pair for g = 1, computing ∆(M ) = {R 1 , R 2 } and B 1 a as we did in Example 6.14.…”

“…Of course, not every true-rank pair is a BC pair and, in the process of searching for new true-rank pairs, by means of Grünbaum's approach [14], one obtains families of examples, see Example 6.15. One of our goals is to give true rank r pairs and important contributions were made by Grinevich [13], Mokhov [27,28,29,30], Mironov [25], Davletshina and Shamaev [9], Davletshina and Mironov [8], Mironov and Zheglov [26,46], Oganesyan [34,35,36], Pogorelov and Zheglov [37]. To check our results we constructed new true rank 2 pairs, by means of non self-adjoint operators of order 4 with genus 2 spectral curves, see Examples 3.2 and 6.14.…”

Explicit Characterization of Some Commuting Differential Operators of Rank 2

“…Operators L 1 and L 2 commute with operators of order 4g+2 (see [12], [13]). The spectral curves of these operators have the form w 2 = z 2m+1 + a 2m z 2m + ... + a 1 z + a 0 .…”

“…The spectral curves of these operators have the form w 2 = z 2m+1 + a 2m z 2m + ... + a 1 z + a 0 . Moreover, the following theorems are proved in [12], [13].…”