Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

TL;DR

This paper studies how spectral invariants of hyperbolic Riemann surfaces behave under elliptic degeneration, showing convergence of some invariants and divergence of others, with regularization techniques applied.

Contribution

It provides a detailed analysis of spectral invariant behavior during elliptic degeneration, including convergence, divergence, and regularization methods, extending previous spectral theory results.

Findings

Selberg zeta function converges to the limit surface

Small eigenvalues and eigenfunctions converge

Spectral zeta function diverges but can be regularized

Abstract

This is the second in a series of two articles where we study various aspects of the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration. In the first article, we investigate the asymptotics of the trace of the heat kernel both near zero and infinity and we show the convergence of small eigenvalues and corresponding eigenfunctions. Having obtained necessary bounds for the trace, this second article presents the behavior of several spectral invariants. Some of these invariants, such as the Selberg zeta function and the spectral counting functions associated to small eigenvalues below 1/4, converge to their respective counterparts on the limiting surface. Other spectral invariants, such as the spectral zeta function and the logarithm of the determinant of the Laplacian diverge. In these latter cases, we identify diverging terms and remove their contributions, thus regularizing convergence of these spectral invariants. Our study is motivated by a result from \cite{He 83}, which D. Hejhal attributes to A. Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we obtain a quantitative result to Selberg's remark.