  Schr\'{o}dinger's equation with distributional $\delta$, or $\delta'$
potentials has been well studied in the past. There are challenges in
simultaneously addressing some of the inherent issues of the system: The
functional operator cannot exist entirely within the standard $L^2$ Hilbert
spaces. On differentiable manifolds, the domain of the free kinetic energy
operator is in the space of harmonic forms. Locally, by the Hodge decomposition
theorem and the standard distributional calculus, the space of functionals of a
$\delta$ or $\delta'$ potential must be orthogonal to the free kinetic energy
operator. Restricting to semicontinuous topologies presents opportunities to
address these, and other issues. We develop, in great detail, a formalism of
Banach spaces with semicontinuous topologies, and their properties are
extensively defined and studied. For $C(\overline{\mathbb{R}})$ functions, the
spaces are indistinguishable. The semicontinuous analogs of the $L^p$ spaces,
are nontrivial and result in a dense topologically continuous embedding of the
semicontinuous $L^p$ spaces into the semicontinuous $C(\overline{\mathbb{R}})$
spaces. Here, certain classes of distributions may be inverted in terms of
their primitive functions. Also many operators are inherently self adjoint. We
define equivalence relations between the cohomology classes of distributions
and derivatives of their associated primitives on local sections of
$\overline{\mathbb{R}}$. Here Hamilton's equations are canonical, and define a
connection on the fibers of the base space. Semicontinuity provides a
resolution to the above domain and interaction problems, and easily integrable
Feynman functional. We arrive at a compatible domain which is Krein
($\mathfrak{H}$) over disjoint components of $\overline{\mathbb{R}}$. The
subspaces of $\mathfrak{H}$ are isomorphic to the semicontinuous Hilbert spaces
of the Hamiltonian.
