Singularities of the wave trace for the Friedlander model

TL;DR

This paper investigates the singularities of the wave trace in the Friedlander model, demonstrating that, similar to the unit disk case, the wave trace exhibits smooth behavior on certain intervals despite complex singularities elsewhere.

Contribution

It extends the analysis of wave trace singularities to the Friedlander model, providing a simpler and more transparent proof of smoothness in specific intervals.

Findings

Wave trace is smooth on certain intervals despite singularities

Friedlander model analysis parallels the unit disk case

Simpler proof of wave trace regularity

Abstract

In a recent preprint, we showed that for the Dirichlet Laplacian $\Delta$ on the unit disk, the wave trace ${Tr}(e^{it\sqrt{\Delta}})$, which has complicated singularities on $2\pi - \epsilon < t < 2\pi$, is, on the interval $2\pi < t < 2\pi + \epsilon$, the restriction to this interval of a $C^\infty$ function on its closure. In this paper we prove the analogue of this somewhat counter-intuitive result for the Friedlander model. The proof for the Friedlander model is simpler and more transparent than in the case of the unit disk.