Aguilar-Balslev-Combes theorem for the Laplacian on a manifold with an axial analytic asymptotically cylindrical end

TL;DR

This paper extends the Aguilar-Balslev-Combes theorem to the Laplacian on manifolds with asymptotically cylindrical ends, allowing for slow metric convergence and defining resonances as non-real eigenvalues of deformed operators.

Contribution

It develops complex scaling for such manifolds under minimal analyticity assumptions and characterizes resonances as poles of the meromorphically continued resolvent.

Findings

Resonances are identified with poles of the continued resolvent.

The Laplacian has no singular continuous spectrum.

Eigenvalues can only accumulate at thresholds.

Abstract

We develop the complex scaling for a manifold with an asymptotically cylindrical end under an assumption on the analyticity of the metric with respect to the axial coordinate of the end. We allow for arbitrarily slow convergence of the metric to its limit at infinity, and prove a variant of the Aguilar-Balslev-Combes theorem for the Laplacian $\Delta$ on functions. In the case of a manifold with (noncompact) boundary it is either the Dirichlet or the Neumann Laplacian. We introduce resonances as the discrete non-real eigenvalues of non-selfadjoint operators, obtained as deformations of the Laplacian by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((\Delta-\mu)^{-1}F, G)$ meromorphic continuation in $\mu$ across the essential spectrum of $\Delta$, where $F$ and $G$ are elements of an explicitly given set of analytic vectors. The Laplacian has no singular continuous spectrum, the eigenvalues can accumulate only at thresholds.