Stark Hamiltonians with Hypersurface-Supported $\delta$-Interactions: Self-Adjoint Realization and Boundary Resolvent Formula

We study Stark Hamiltonians with a $\delta$-interaction supported on a compact hypersurface in ${\mathbb{R}}^d$. Let $\mathrm{\Sigma }$ be a compact Lipschitz hypersurface and let $\alpha \in L^{\mathrm{\infty }}(\mathrm{\Sigma };\mathbb{R})$. We define the operator $H_{F,\alpha }$ as a self–adjoint realization of the formal Hamiltonian $H_{F,0}+\alpha {\delta }_{\mathrm{\Sigma }}$ by imposing transmission conditions across $\mathrm{\Sigma }$. We then derive a boundary resolvent formula which expresses the resolvent of $H_{F,\alpha }$ in terms of the free Stark resolvent and a boundary operator on $\mathrm{\Sigma }$. This reduces the spectral problem to the boundary and shows that the interaction can be treated as a boundary perturbation at the resolvent level. As an application, we prove that for every nonzero electric field the resolvent difference between $H_{F,\alpha }$ and $H_{F,0}$ is compact on $L^2({\mathbb{R}}^d)$. It follows that the essential spectrum of $H_{F,\alpha }$ coincides with $\mathbb{R}$. The argument is based on trace mapping properties for compact Lipschitz hypersurfaces and does not rely on translation invariance of the background operator.