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This article is cited in 13 scientific papers (total in 13 papers)
Invertible Darboux Transformations
Ekaterina Shemyakova Department of Mathematics, SUNY at New Paltz, 1 Hawk Dr. New Paltz, NY 12561, USA
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.
Keywords:
Darboux transformations; Laplace transformations; 2D Schrödinger operator; invertible Darboux transformations.
Received: October 1, 2012; in final form January 1, 2013; Published online January 4, 2013
Citation:
Ekaterina Shemyakova, “Invertible Darboux Transformations”, SIGMA, 9 (2013), 002, 10 pp.
Linking options:
https://www.mathnet.ru/eng/sigma785 https://www.mathnet.ru/eng/sigma/v9/p2
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