  Elegant 'microlocal' methods have long since been extensively developed for
the analysis of conventional Schroedinger eigenvalue problems. For technical
reasons though these methods have not heretofore been applicable to quantum
field theories. In this article however we initiate a 'Euclidean signature
semi-classical' program to extend the scope of these analytical techniques to
encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1
dimensions. The basic microlocal approach entails, first of all, the solution
of a single, nonlinear equation of Hamilton-Jacobi type followed by the
integration (for both ground and excited states) of a sequence of linear
'transport' equations along the 'flow' generated by the 'fundamental solution'
to the aforementioned Hamilton-Jacobi equation. Using a combination of the
direct method of the calculus of variations, elliptic regularity theory and the
Banach space version of the implicit function theorem we establish, in a
suitable function space setting, the existence, uniqueness and global
regularity of this needed 'fundamental solution' to the relevant, Euclidean
signature Hamilton-Jacobi equation for the systems under study. Our methods are
applicable to (massive) scalar fields with polynomial self-interactions of
renormalizable type. They can, as we shall show elsewhere, also be applied to
Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.
