A nonvariational form of the acoustic single layer potential

We consider a bounded open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha }$ for some $\alpha \in ]0,1[$ and the space $V^{-1,\alpha }(\mathrm{\partial }\mathrm{\Omega })$ of (distributional) normal derivatives on the boundary of $\alpha$-Hölder continuous functions in $\mathrm{\Omega }$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,\alpha }(\bar{\mathrm{\Omega }})$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $\mathrm{\Omega }$ with boundary data in $V^{-1,\alpha }(\mathrm{\partial }\mathrm{\Omega })$ and solutions in the space of $\alpha$-Hölder continuous functions in $\mathrm{\Omega }$ that have Laplace operator in $C^{-1,\alpha }(\bar{\mathrm{\Omega }})$, $\mathit{\text{i.e.}}$, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting.