Complex scaling for the Dirichlet Laplacian in a domain with asymptotically cylindrical end

TL;DR

This paper introduces a complex scaling method for analyzing the Dirichlet Laplacian in domains with asymptotically cylindrical ends, enabling the identification of resonances as eigenvalues of deformed operators.

Contribution

It develops a novel complex scaling approach for the Dirichlet Laplacian in specific geometries, defining resonances via non-selfadjoint operator eigenvalues.

Findings

Resonances correspond to poles of the meromorphically continued resolvent.

The Dirichlet Laplacian has no singular continuous spectrum.

Eigenvalues can only accumulate at spectral thresholds.

Abstract

We develop the complex scaling method for the Dirichlet Laplacian in a domain with asymptotically cylindrical end. We define resonances as discrete eigenvalues of non-selfadjoint operators, obtained as deformations of the selfadjoint Dirichlet Laplacian $\Delta$ 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 partial analytic vectors. It turns out that the Dirichlet Laplacian has no singular continuous spectrum, and its eigenvalues can accumulate only at threshold values of the spectral parameter.