2016/03/04 by Daniel Garbin, Garbin, Daniel, Jay Jorgenson +1
Mathematics · Social Sciences · #Analytic and geometric function theory #European Linguistics and Anthropology #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1603.01494
openalex publication_date 2016/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the second in a series of two articles where we study various aspects\nof the spectral theory associated to families of hyperbolic Riemann surfaces\nobtained through elliptic degeneration. In the first article, we investigate\nthe asymptotics of the trace of the heat kernel both near zero and infinity and\nwe show the convergence of small eigenvalues and corresponding eigenfunctions.\nHaving obtained necessary bounds for the trace, this second article presents\nthe behavior of several spectral invariants. Some of these invariants, such as\nthe Selberg zeta function and the spectral counting functions associated to\nsmall eigenvalues below 1/4, converge to their respective counterparts on the\nlimiting surface. Other spectral invariants, such as the spectral zeta function\nand the logarithm of the determinant of the Laplacian diverge. In these latter\ncases, we identify diverging terms and remove their contributions, thus\nregularizing convergence of these spectral invariants. Our study is motivated\nby a result from citeHe 83, which D. Hejhal attributes to A. Selberg,\nproving spectral accumulation for the family of Hecke triangle groups. In this\narticle, we obtain a quantitative result to Selberg's remark.\n