Absence of singular continuous spectrum for some geometric Laplacians

TL;DR

This paper demonstrates the absence of singular continuous spectrum for certain geometric Laplacians on manifolds with corners, using adapted spectral analysis techniques including analytic dilation.

Contribution

It introduces a novel adaptation of analytic dilation to geometric Laplacians on manifolds with corners, providing new spectral insights.

Findings

Absence of singular continuous spectrum for specific geometric Laplacians

Description of pure point spectrum behavior based on geometry

Development of a theory of quantum resonances

Abstract

We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators, a description of the behavior of its pure point spectrum in terms of the underlying geometry, and a theory of quantum resonances.