Dynamical generalized functions and the multiplication problem

Abstract

1. In this paper, we consider the problem ofmultiplication of a generalized function by a discontinuous function. The need in such an operation appears in the study of ordinary differential equations with generalized functions [1–8]; it can be viewed as a particular case of a more general operation of multiplication of two generalized functions [9–11] the latter operation arises, in particular, in applications to the problems of quantum mechanics [12, 13]. As is known [14, 15], in the classical space D′, it is impossible to define a continuous operation of multiplication of generalized functions. It is also impossible to define a continuous partial operation of multiplication of a generalized function from D′ by a discontinuous function [14, 15]. A continuous operation of elements of D′ is defined in the algebra of Colombeau generalized functions [9], which contains D′ as a subspace. In the algebra of Colombeau generalized functions, there exists a product of any two elements of D′, although, in the general case, this product is a Colombeau generalized function and does not belong to D′ [9]. In particular, the product of the unit function θτ discontinuous at τ and the delta-function δτ does not belong to D′. This leads to a series of inconveniences in the study of differential equations with generalized functions containing the product of a generalized function and a discontinuous function. The main inconvenience is in the fact that the solution is not an ordinary discontinuous function but a Colombeau generalized function, which makes it difficult to give its physical interpretation [9, 16]. In this paper, we construct a space of dynamical generalized functions T ′ in which a continuous and associative operation of multiplication of a generalized function by a discontinuous function is defined. The definitions of the product of the unit function θτ and the classical delta-function δτ ∈ D′ given in [4, 11, 17, 18] (which, generally speaking, are neither associative, nor continuous) are particular cases of the multiplication in the space T ′.