Seminar talk, 20 December 2017

From Geometry of Differential Equations
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Speaker: Ekaterina Shemyakova

Title: Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator

Formal sums of densities of various weights under the natural multiplication form a commutative algebra associated to a given (super)manifold. The algebra of densities was introduced by Khudaverdian and Voronov in the works on odd Laplace operators. It extends the algebra of functions and its elements themself can be interpreted as functions on "extended manifolds" of dimension greater by one. It's very natural to consider different objects related to this algebra, for example, differential operators. In the talk we discuss the factorization problem for such operators. This is a standard problem in differential algebra. It is known that the factorizations of one-dimensional, that is, ordinary, and multidimensional, that is, partial, differential operators are fundamentally different. We shall discuss the differential operators in the algebra of densities on the line. On one hand, this is a one-dimensional situation; but due to special nature of the algebra on which the operators act, this in a certain sense is in between the 1- and 2-dimensional cases. We show that unlike the familiar case, the operators in question are not always factorizable and establish a criterion of factorizabily for the generalized Sturm-Liouville operator on the algebra of densities. As a matter of interest, the criterion obtained can be formulated in terms of solution of the classical Sturm-Liouville equation. By the way, we establish the possibility of an incomplete factorization of the generalized Sturm-Liouville operator.

The talk is based on paper arXiv:1710.09542, joint with Theodore Voronov.