Spectrum and Resonances of Stark Hamiltonians with $δ$--Shell Potentials

Abstract: We study the Stark Hamiltonian with a $\delta$--interaction supported on a hypersphere, with boundary strength $\alpha\in L^\infty(S_a)$. Using the framework of quasi boundary triples, we construct self--adjoint realizations and derive a Krein--type resolvent formula. For $F\neq0$ we prove $\sigma(H_{F,\alpha})=\sigma_{\mathrm{ac}}(H_{F,\alpha})=\mathbb R$ by transferring a strict Mourre estimate from the free Stark operator and by the compactness of the boundary coupling. Resonances are defined by complex distortion in the field direction and characterized as zeros of a boundary regularized Fredholm determinant. In $d=3$ and small $F$, bound states at $F=0$ continue to resonances with widths of order $\exp(-c/F)$, governed by an Agmon--type distance. A brief comparison with the free Stark case is also given.