Invertible Darboux Transformations

Symmetry Integrability and Geometry-Methods and Applications

article

Ekaterina Shemyakova¹ 

[1] Department of Mathematics

关键词: Darboux transformations;    Laplace transformations;    2D Schr¨odinger operator;    invertible Darboux transformations;   

DOI  :  10.3842/SIGMA.2013.002

来源: National Academy of Science of Ukraine

【 摘 要 】

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.

【 授权许可】

Unknown