Some Remarks on Scattering Resonances in Even Dimensional Euclidean Scattering

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 study the multiplicities of the poles of the continued scattering matrix on each sheet of Λ and relate these to the multiplicities of the poles of the resolvent on each sheet. 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 there are no “purely imaginary” resonances on any sheet of Λ for Schrödinger operators with potentials 0 ≤ V ∈ L0 (R) in even dimensions.