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 $\Sigma$ be a compact Lipschitz hypersurface and let $\alpha\in L^\infty(\Sigma;\mathbb R)$. We define the operator $H_{F,\alpha}$ as a self--adjoint realization of the formal Hamiltonian $H_{F,0}+\alpha\delta_\Sigma$ by imposing transmission conditions across $\Sigma$. 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 $\Sigma$. 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.