Commuting differential operators

2010 ◽  
Vol 162 (3) ◽  
pp. 276-285 ◽  
Author(s):  
A. B. Shabat ◽  
Z. S. Elkanova
Keyword(s):  
Differential Operators ◽  
Commuting Differential Operators

Burchnall–Chaundy theory, Ore extensions and σ-differential operators

2014 ◽  
Vol 13 (07) ◽  
pp. 1450049
Author(s):  
Daniel Larsson
Keyword(s):  
Algebraic Curve ◽  
Differential Operators ◽  
Classical Theorem ◽  
Precise Statement ◽  
Commuting Differential Operators ◽  
Ore Extensions

A classical theorem of J. L. Burchnall and T. W. Chaundy shows that two commuting differential operators P and Q give rise, via a differential resultant, to a complex algebraic curve with equation F (x, y) = 0, such that formally inserting P and Q for x and y in F (x, y) , gives identically zero. In addition, the points on this curve have coordinates which are exactly the eigenvalues associated with the operators P and Q (see the Introduction for a more precise statement). In this paper, we prove a generalization of this result using resultants in Ore extensions.

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