  In this paper we consider the Cauchy problem for multidimensional elliptic
equations in a cylindrical domain. The method of spectral expansion in
eigenfunctions of the Cauchy problem for equations with deviating argument
establishes a criterion of the strong solvability of the considered elliptic
Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy
problem is equivalent to the existence of an isolated point of the continuous
spectrum for a self-adjoint operator with deviating argument.
