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

Semicontinuous Banach spaces for Schr\"odinger's Eq. with Dirac-$\delta'$ potential