Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

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 [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. * The first author acknowledges support from a PSC-CUNY grant. The second author acknowledges support from grants from the NSF and PSC-CUNY.

The parametrix construction of the heat kernel on a graph \footnote{Keywords: heat kernels, graphs, parametrix, Bessel functions.}

“…In doing so, one immediately obtains a classical theta inversion formula, which itself is logically equivalent to the one-variable Poisson summation formula. A similar argument in the setting of compact hyperbolic Riemann surfaces yields a quick proof of the Selberg trace formula; see Remark 3.3 of [GJ18]. In the case of a heat kernel on a discrete circle with N points, then similar considerations in [KN06] lead to łheat kernel proofsž of I-Bessel identities.…”