Abstract
We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in Adachi et al. (JDE 268: 5179–5206, 2020; Stationary scattering theory for 1-body Stark operators). The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the method of Isozaki-Kitada (Tokyo Univ. 35: 81–107, 1985) used for studying the scattering matrix for one-body Schrödinger operators without an external potential. It is more flexible and more informative than the more standard method used previously by Kvitsinsky-Kostrykin (Teoret. Mat. Fiz. 75(3): 416-430, 1988) for computing the leading order singularities of the kernel of the scattering matrix in the case of a constant external field (the Stark case). Our approach relies on Sommerfeld’s uniqueness result in Besov spaces, microlocal analysis as well as on classical phase space constructions.
| Original language | English |
|---|---|
| Journal | Annales Henri Poincare |
| Volume | 23 |
| Issue | 2 |
| Pages (from-to) | 513-548 |
| Number of pages | 36 |
| ISSN | 1424-0637 |
| DOIs | |
| Publication status | Published - 2022 |