Noncommutative bispectral Darboux transformations

Geiger, Joel; Horozov, Emil; Yakimov, Milen

doi:10.1090/tran/6950

Noncommutative bispectral Darboux transformations
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by Joel Geiger, Emil Horozov and Milen Yakimov PDF

Trans. Amer. Math. Soc. 369 (2017), 5889-5919 Request permission

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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