Darboux Transformations for Factorable Laplace Operators

摘要

Factorable Laplace operators of the form L = (б)_x(б)_y + a(б)_x + b(б)_y + c, where the coefficients a, b, c are not necessarily constants, are considered. For these operators, the Darboux transformations L(M→) L_1, M ∈ K[(б)_X], defined by the intertwining relation NL = L_1M are considered. It is shown that only the following cases are possible: either (1) M ⌒ ker(б)_x + b = {0} and L_1 is also factorable or (2) M ⌒ ker(б)_x + b contains a nonzero element. We prove that, in both cases, the Darboux transformation can be represented as a product of first-order Darboux transformations. For case (2), the proof is based on the fact that the Darboux transformation of operator L can be reduced to Darboux transformations of first-order operators.