1304.7063 (Ekaterina Shemyakova)

Factorization of Darboux Transformations of Arbitrary Order for
Two-dimensional Schrödinger operator    [PDF]

Ekaterina Shemyakova

We prove that a Darboux transformation of arbitrary order $d$ (and in a normalized form) for two-dimensional Schr\"odinger operator can be can be factored into Darboux transformations of order 1. Even for the special case of Darboux transformations of order 2 this problem is hard. For this case a rather beautiful proof (based on the ivariantization and the regularized moving frames \cite{olver2011diff_inv_algebras,OP:05,Mansf_book}) of the statement is suggested in \cite{shem:darboux2}. The analogous problem for one-dimensional Schr\"odinger operator was proved in four steps \cite{veselov:shabat:93,shabat1995,bagrov_samsonov_1995,clouds:bagrov:samsonov:97}. In this case the factorization is not unique and different factorizations imply discrete symmetries related to the Yang-Baxter maps \cite{adler1993,Veselov2003}. The main result of the present paper implies that a Darboux transformation of arbitrary form and of arbitrary order $d$ is invertible only in the case where it is a Laplace chain.

View original: http://arxiv.org/abs/1304.7063