Necessary and Sufficient Conditions for First Order Differential Operators to be Associated with a Disturbed Dirac Operator in Quaternionic Analysis

Abstract

Recently the initial value problem

$$\left.\begin{array}{rl}\partial_{t}u & = \mathcal{L}u := \sum\limits_{i=1}^3 A^{(i)}(t, x)\partial_{x_i}u + B(t, x)u + C(t, x) \\ u (0, x) & = u_{0}(x) \end{array}\right.$$

has been solved uniquely by N. Q. Hung (Adv. appl. Clifford alg., Vol. 22, Issue 4 (2012), pp. 1061-1068) using the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) in the space of generalized regular functions in the sense of quaternionic analysis satisfying the equation ${\mathcal{D}_{\alpha}u = 0}$, where

$$\mathcal{D}_{\alpha}u := \mathcal{D}u + \alpha u, \quad \alpha \in \mathbb{R},$$

and ${\mathcal{D} = \sum\limits_{j=1}^{3}ej \partial_{x_j}}$ is the Dirac operator, x = (x1, x2, x3) is the space like variable running in a bounded domain in ${\mathbb{R}^{3}}$, and t is the time. The author has proven only sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}\alpha}$, i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\alpha}u = 0}$ into solutions of the same equation for fixedly chosen t.

In the present paper we prove necessary and sufficient conditions for the underlined operators to be associated. This criterion makes it possible to construct all linear operators ${\mathcal{L}}$ for which the initial value problem with an arbitrary initial generalized regular function is always solvable.