On the Dirac spectrum on degenerating Riemannian surfaces

TL;DR

This paper analyzes how the spectrum of the Dirac operator on degenerating Riemannian surfaces behaves as a simple closed geodesic shrinks, using a specialized pseudodifferential calculus to handle the non-compact limit surface.

Contribution

It introduces an adapted pseudodifferential calculus to study Dirac spectra on degenerating surfaces with non-trivial spin structures, addressing non-compactness issues.

Findings

Spectral projectors remain smooth during degeneration

Established $t^2 \log t$ regularity for the cusp-surgery resolvent trace

Analyzed Dirac operator behavior on limit non-compact surfaces

Abstract

We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length $t$ of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated b-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. We obtain smoothness of the spectral projectors, and $t^2 \log t$ regularity for the cusp-surgery trace of the relative resolvent in the degeneracy process as $t \searrow 0$.