ON THE ASYMPTOTIC BEHAVIOR OF

COUNTING FUNCTIONS ASSOCIATED TO

DEGENERATING HYPERBOLIC RIEMANN SURFACES

Jonathan Huntley Jay Jorgenson Rolf Lundelius

Abstract. We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if M is compact, is defined for w ≥ 0 and T > 0 by

N_{M,w}(T) = ∑_{λ_{n}≤T}(T − λ_{n})^{w}

where {λ_{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 the weighted counting function is defined via the inverse Laplace transform. Now let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. As an example of our results, we have the following theorem: There is an explicitly defined function G_{ℓ,w}(T) which depends solely on ℓ, w, and T such that for w > 3/2 and T > 0, we have

N_{M_{ℓ},w}(T) = G_{ℓ,w}(T) + N_{M_{0},w}(T) + o(1)

for ℓ → 0. We also consider the setting when w < 3/2, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume.

Typeset by AMS-TEX