Equatios on a superspace

A. Yu. Khrennikov

Izvestiya. Mathematics , Tome 36 (1991) no. 3, pp. 597-627

Résumé

The construction of a general theory of partial differential equations on a superspace is continued in the framework of functional superanalysis. Superanalogues of the spaces $\mathscr S(\mathbf R^n)$ and $\mathscr D(\mathbf R^n)$ of generalized functions are introduced; a theorem is proved on the existence of a fundamental solution for linear differential equations with constant coefficients on a superspace. In contrast to the scalar case, there exist differential operators not having fundamental solutions. Formulas are obtained for the fundamental solutions of the Laplace operator, the heat conduction operator, the Schrödinger operator, the d'Alembert operator, and the Helmholtz operator on a superspace. There is a discussion of the role of the nilpotence condition for even ghosts in a commutative superalgebra in the construction of a theory of generalized functions.