Spectral theory for 1-body Stark operators

Abstract

We investigate spectral theory for a one-body Stark Hamiltonian under minimum regularity and decay conditions on the potential (actually allowing sub-linear growth at infinity). Our results include Rellich's theorem, the limiting absorption principle, radiation condition bounds and Sommerfeld's uniqueness, and most of these are stated and proved in sharp form employing Besov-type spaces. For the proofs we adopt a commutator scheme by Ito–Skibsted [13]. A feature of the paper is a special choice of an escape function related to parabolic coordinates, which conforms well with classical mechanics for the Stark Hamiltonian. The whole setting of the paper, such as the conjugate operator and the Besov-type spaces, is generated by this single escape function. We apply our results in the sequel paper [5].