I am somehow intrigued by a statement I found in Glimm and Jaffe, Quantum Physics: a Functional Integral Point of View, 2nd edition, section 7.8.
The general idea, given a region $\Lambda$ of $\mathbf{R}^d$, is to relate the Laplace operator $\Delta_{\partial\Lambda}$ with Dirichlet boundary conditions on $\partial\Lambda$ to the Laplace operator $\Delta$ on $\mathbf{R}^d$ by a limiting process. Namely, defining: $$C(\lambda) \equiv \left(-\Delta + m^2 + \lambda\chi_{\Lambda^c}\right)^{-1}$$ where $m, \lambda > 0$, one first proves that $C(\lambda)$ converges strongly and decreasingly to $C \equiv \chi_{\Lambda}\left(-\Delta_{\partial\Lambda} + m^2\right)^{-1}\chi_{\Lambda}$. Here, $\chi_E$ denotes the multiplication operator by the characteristic function of the set $E$. Looking at $C$ as an operator acting on $L^2(\Lambda)$, this is shown to imply $C^{-1} = -\Delta_{\partial\Lambda} + m^2$.
Now, here is what is perplexing me: with these notations set up, the authors claim that for all $t > 0$ and $x, y \in \mathbf{R}^d$, the kernel $e^{-tC(\lambda)^{-1}}(x, y)$ tends to $\chi_{\Lambda}e^{-t\left(-\Delta_{\partial\Lambda} + m^2\right)}\chi_{\Lambda}$ as $\lambda \to \infty$. I find it a bit fast... Could anyone fill in the gaps?