that $H_{t}$ and ${\partial}_{t} H_{t}$ are bounded, self-adjoint operators on $L^{2}$.

Theorem \[thm:heatkernelmap\] implies the essential self-adjointness statement; indeed the fundamental solution $H_t$ directly gives a sequence (indeed a path) of sections on $\mathcal{D}_{\min}$ which approaches a given form in $\mathcal{D}_{\max}$. Namely, $$\label{eq:obvious-approximation}
  \alpha \in \mathcal{D}_{\max} \implies H_t \alpha \to \alpha \mbox{
    in } \mathcal{D}_{\min} \mbox{ as } t \downarrow 0.$$ As we see now, the proof of this is straightforward functional analysis given the conclusions of Theorem \[thm:heatkernelmap\].

The proof has nothing to do with the fine structure of incomplete cusp edge spaces, it depends only on the soft properties of the fundamental solution $H$ in Theorem \[thm:heatkernelmap\]. To emphasize this, let $(M,
g)$ be any Riemannian manifold and $P$ a differential operator of order $2$ acting on sections of a vector bundle $E$ with hermitian metric $G$, such that $P$ is symmetric on $L^{2}(M; E)$. For $t > 0$, let $H_{t}$ be a smooth section of $\operatorname{End}(E) \to M
\times M$ which depends smoothly on $t$ and satisfies $$\label{eq:1}
  ({\partial}_{t} + P) H_{t} = 0 \quad \mbox{ and } \quad \lim_{t \to 0} H_{t} = \operatorname{Id},$$ where the latter limit holds in the strong topology on $L^{2}$, and 