2013/07/22 by T. J. Christiansen, Christiansen, T. J., Peter D. Hislop +2 · 1 citation
Mathematics · Physics and Astronomy · #35P25 #81U05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:35P25 #msc:81U05
paper · pdf · doi:10.48550/arxiv.1307.5822
27 pages
arxiv created 2013/07/22 · openalex publication_date 2013/07/22 · arxiv updated 2013/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions d ≥ 2. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto Λ, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the mth sheet of Λ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a number of results about "pure imaginary" resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no "purely imaginary" resonances on any sheet of Λ for Schrödinger operators with potentials 0 ≤ V ∈ L0^∞ (\Rd).