XXXII Workshop on Geometric Methods in Physics 30.06-6.07.2013

Ekaterina Shemyakova

Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schroedinger operator

We prove that a Darboux transformation of arbitrary order d for two-dimensional Schroedinger operator
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 we have found earlier a rather beautiful proof based on the invariantization (we used regularized moving frames due to Olver and Pohjanpelto).

The analogous statement for one-dimensional Schroedinger operator was proved in four steps (Shabat, Veselov and Bagrov, Samsonov). In this case the factorization is not
unique, and different factorizations imply discrete symmetries related to the Yang-Baxter maps (Adler and Veselov).

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