Explicit Characterization of Some Commuting Differential Operators of Rank 2

Commuting Differential Operators of Rank 2 with Rational Coefficients

“…From system (17) we get So, we obtain that system (17) always has solution. Now let us consider system (18).…”

“…x +u(x) and M 2 = L 2g+1 4 + β 2g L 2g 4 + ... + β 0 were studied in [17]. Mironov and Zheglov (see [11]) proved that for arbitrary integer m and arbitrary spectral curve Γ given by equation…”

Matrix Commuting Differential Operators of Rank 2 and Arbitrary Genus

“…0 , α 2 = 64n 4 − 4n 2 , n ∈ N commutes with a differential operator (14), of order 4n, where g = 2n. The order of operator M equals 4n.…”

“…In [11] Mironov developed theory of self-adjoint scalar operators of rank 2 and found examples of commuting scalar operators of rank 2 and arbitrary genus. Using Mironov's method many examples of scalar commuting operators of rank 2 and arbitrary genus were found (see [12], [13], [14], [15], [16]).…”

Commuting Ordinary Differential Operators and the Dixmier Test

“…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.…”