Algebra of three-dimensional generalized functions

The method proposed by the author in an earlier paper [1] is used to construct the associative algebra $\mathscr{A}$$(3)$, which is equipped with involution and differentiation, for generalized functions of three variables that at a~fixed point can have singularities of the type~$\delta(\mathbf{r})$, $r^{-1}$, $r^{-2}$ and their derivatives. In complete analogy with the one-dimensional algebra of~[1], the elements of the algebra~$\mathscr{A}$$(3)$ form in conjunction with the differentiation operator an algebra of local operators of quantum theory with indefinite metric and with state vectors that are also generalized functions. It is noted that one can go over to smaller spaces of state vectors and obtain three-dimensional Schrödinger equations with strongly singular potentials and positive metric.