Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 002, 10 pages      arXiv:1210.0803      http://dx.doi.org/10.3842/SIGMA.2013.002
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Invertible Darboux Transformations

Ekaterina Shemyakova
Department of Mathematics, SUNY at New Paltz, 1 Hawk Dr. New Paltz, NY 12561, USA

Received October 01, 2012, in final form January 01, 2013; Published online January 04, 2013

Abstract
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.

Key words: Darboux transformations; Laplace transformations; 2D Schrödinger operator; invertible Darboux transformations.

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