[Paper Review] Some remarks on resonances in even-dimensional Euclidean scattering

T. J. Christiansen, Peter D. Hislop|arXiv (Cornell University)|Jul 22, 2013

Spectral Theory in Mathematical Physics5 references6 citations

TL;DR

This paper establishes that in even-dimensional Euclidean scattering, Schrödinger operators with non-negative or non-positive compactly supported potentials have no purely imaginary resonances on any sheet of the logarithmic Riemann surface $Λ$, except possibly finitely many for negative potentials, which are tied to negative eigenvalues. The results resolve a contrast with the odd-dimensional case and correct a subtle identity in scattering theory relevant to resonance symmetries.

ABSTRACT

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. 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 $m$th 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 $0 \leq V \in L_0^\infty (\R^d)$.

Motivation & Objective

• To clarify the relationship between poles of the scattering matrix and the resolvent in even-dimensional Euclidean scattering.
• To investigate the existence of purely imaginary resonances in even dimensions, contrasting with the well-known abundance in odd dimensions.
• To correct a subtle error in a standard identity from [27] concerning symmetries of the scattering matrix.
• To show that poles of the scattering matrix on the $m$th sheet of $\Lambda$ correspond to zeros of a scalar function on the physical sheet.
• To establish the absence of purely imaginary resonances for non-negative potentials and a finite upper bound for negative potentials in even dimensions.

Proposed method

• Use the black box scattering framework of Sjöstrand and Zworski to define resonances via meromorphic continuation of the resolvent and scattering matrix to the logarithmic Riemann surface $\Lambda$.
• Apply a corrected identity for the free resolvent's analytic continuation across sheets: $R_0(e^{im\pi}\lambda) = R_0(\lambda) + imT(\lambda)$, with $T(\lambda)$ having a specific integral kernel.
• Analyze the resolvent formula $|V|^{1/2}R_V(\lambda)|V|^{1/2}(I + (\operatorname{sgn}V)|V|^{1/2}R_0(\lambda)|V|^{1/2}) = |V|^{1/2}R_0(\lambda)|V|^{1/2}$ to locate resonances as zeros of $I + K(\lambda)$.
• Study the operator $K(ie^{im\pi}\sigma)$ for purely imaginary $\lambda = i\sigma$ to determine conditions for zeros on the $m$th sheet.
• Use the self-adjointness of $\chi T(i\sigma)\chi$ for even $d$ to show that the perturbation term is skew-adjoint, enabling spectral analysis.
• Relate the existence of purely imaginary resonances to the presence of negative eigenvalues of $-\Delta + V$ when $V \leq 0$.

Experimental results

Research questions

• RQ1 Do purely imaginary resonances exist for Schrödinger operators with non-negative potentials in even dimensions?
• RQ2 How do the poles of the scattering matrix on the $m$th sheet of $\Lambda$ relate to a scalar function on the physical sheet?
• RQ3 What is the correct form of the symmetry identity for the scattering matrix in even dimensions, and how does it affect the analysis of purely imaginary resonances?
• RQ4 Can the number of purely imaginary resonances be bounded for negative potentials in even dimensions?
• RQ5 How does the absence of purely imaginary resonances in even dimensions contrast with the infinite family of such resonances in odd dimensions?

Key findings

• For $V \geq 0$ and even $d \geq 2$, there are no purely imaginary resonances on any sheet $\Lambda_m$, $m \in \mathbb{Z}$.
• For $V \leq 0$ and even $d \geq 2$, there are at most $N_V < \infty$ purely imaginary resonances on each sheet $\Lambda_m$ for $m \neq 0$, where $N_V$ is the number of negative eigenvalues of $-\Delta + V$.
• The poles of the scattering matrix on the $m$th sheet $\Lambda_m$ correspond exactly to the zeros of a scalar function defined on the physical sheet $\Lambda_0$, establishing a direct correspondence.
• The correction to the identity in [27] is essential: the difference $R_0(e^{im\pi}\lambda) - R_0(\lambda)$ is not self-adjoint in even dimensions, but rather involves a skew-adjoint term $imT(\lambda)$.
• The operator $\chi T(i\sigma)\chi$ is self-adjoint for even $d$, which is crucial for proving the absence of resonances in the non-negative case.
• For $V \leq 0$, a purely imaginary resonance at $\lambda = e^{i(m\pi + \pi/2)}\sigma$ implies $-\sigma^2$ is an eigenvalue of $-\Delta + V$, linking resonances to the discrete spectrum.