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\(\mathcal{D}\)-modules and Darboux transformations. (English) Zbl 0979.13025
Summary: A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of \(\mathcal D\)-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.

MSC:
13N10 Commutative rings of differential operators and their modules
35P05 General topics in linear spectral theory for PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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