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On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

✍ Scribed by Jonathan Huntley; Jay Jorgenson; Rolf Lundelius

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 369 KB
Volume: 149
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1997.3097

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✦ Synopsis

In this article we will study what we call weighted counting functions on hyperbolic Riemann surfaces of finite volume. If M is compact, then we define the weighted counting function for w 0 to be N M, w (T )= :

where [* n ] is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on M. If M is non-compact, then we define the weighted counting function N M, w (T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated to M (see Proposition 5.1 and Remark 5.2). Using the convergence results from [JL2] concerning the regularized heat trace on finite volume hyperbolic Riemann surfaces, we shall prove the following results.

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Convergence of the Normalized Spectral C

Convergence of the Normalized Spectral Counting Function on Degenerating Hyperbolic Riemann Surfaces of Finite Volume

✍ Jay Jorgenson; Rolf Lundelius 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 425 KB

In this paper we study spectral asymptotics of degenerating families of hyperbolic Riemann surfaces, either compact or non-compact but always of finite volume. We prove that the second integral of the spectral counting function has an asymptotic expansion out to o(l), where l is the degeneration par