Seminar talk, 3 October 2018

From Geometry of Differential Equations
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Speaker: Vladimir Chetverikov

Title: Invertible linear ordinary differential operators and their generalizations

Abstract:
We consider invertible linear ordinary differential operators whose inversions are also differential operators. To each such operator one assigns a numerical table. We describe these tables in the elementary geometrical language. The table does not uniquely determine the operator. We present mathematical structures that should be specified for its unique determination.

We say that a linear differential operator is unicellular, if in some bases of the modules the operator is given by an upper triangular matrix that differs from the identity matrix only by the first row. The numerical tables of unicellular operators are of the simplest form. We show that any invertible linear ordinary differential operator is represented as compositions of unicellular ones.

These results are generalized to invertible mappings of filtered modules generated by one differentiation. Invertible linear ordinary differential operators, invertible linear difference operators with periodic coefficients, unimodular matrices, and C-transformations of control systems determine mappings of this type. The possibility of generalization of these results to partial differential operators is also discussed.