The Scattering Matrix and its Meromorphic Continuation in the Stark Effect Case

Abstract

Quantum scattering in the presence of a constant electric field (‘Stark effect’) is considered. It is shown that the scattering matrix has a meromorphic continuation in the energy variable to the entire complex plane as an operator on L²(R ^n-1). The allowed potentials V form a general subclass of potentials that are short-range relative to the free Stark Hamiltonian: Roughly, the potential vanishes at infinity, and admits a decomposition $V = V_\mathcal{A} + V_e$ , where $V_\mathcal{A}$ is analytic in a sector with $V_\mathcal{A} (x) = O(\left\langle {x_{} } \right\rangle ^{ - 1/2 - \varepsilon } )$, and $V_e (x) = O({\text{e}}^{\mu x_1 } )$, for x₁<0 and some μ μ>0. These potentials include the Coulomb potential. The wave operators used to define the scattering matrix are the two Hilbert space wave operators.