
SIGMA 9 (2013), 002, 10 pages arXiv:1210.0803
https://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.
pdf (319 kb)
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