Some remarks on resonances in even-dimensional Euclidean scattering

Abstract:

Black box quantum mechanical scattering on $\mathbb {R}^d$ in even dimensions $d \geq 2$ has many characteristics distinct from the odd-dimensional situation. In this article, we study the scattering matrix in even dimensions and prove several identities which hold for its meromorphic continuation onto $\Lambda$, the Riemann surface of the logarithm function. We prove a theorem relating the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the continued resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $\Lambda$ are determined by the zeros of a scalar function defined on the physical sheet. Although analogs of these results are well known in odd dimension $d$, we are unaware of a reference for all of $\Lambda$ for the even-dimensional case. Our analysis also yields some surprising results about “pure imaginary” resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no “pure imaginary” resonances on any sheet of $\Lambda$ for Schrödinger operators with potentials $0 \leq V \in L_0^\infty (\mathbb {R}^d)$.