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Transactions of the American Mathematical Society

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Noncommutative bispectral Darboux transformations

Authors: Joel Geiger, Emil Horozov and Milen Yakimov
Journal: Trans. Amer. Math. Soc. 369 (2017), 5889-5919
MSC (2010): Primary 37K35; Secondary 16S32, 39A70
Published electronically: February 13, 2017
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Abstract: We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and $ q$-difference operators with values in all noncommutative algebras. All known bispectral Darboux transformations are special cases of the theorem. Using the methods of quasideterminants and the spectral theory of matrix polynomials, we explicitly classify the set of bispectral Darboux transformations from rank one differential operators and Airy operators with values in matrix algebras. These sets generalize the classical Calogero-Moser spaces and Wilson's adelic Grassmannian.

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Additional Information

Joel Geiger
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307

Emil Horozov
Affiliation: Department of Mathematics and Informatics, Sofia University, 5 J. Bourchier Boulevard, Sofia 1126, Bulgaria — and — Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Milen Yakimov
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: The noncommutative bispectral problem, Darboux transformations, matrix rank one bispectral functions, the Airy bispectral function
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: July 11, 2016
Published electronically: February 13, 2017
Additional Notes: The research of the third author was partially supported by NSF grant DMS-1303036 and Louisiana Board of Regents grant Pfund-403
Article copyright: © Copyright 2017 American Mathematical Society

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