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

Jonathan Huntley, Jay Jorgenson and Rolf Lundelius

MSRI #1995-021

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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 $w \geq 0$ and $T > 0$ by $N_{M,w}(T) = \sum\limits_{\lambda_n
\leq T}(T-\lambda_n)^w$ where $\{\lambda_n\}$ 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 $M_{\ell}$ 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
$G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such
that for $w > 3/2$ and $T>0$, we have, for $\ell \to 0$, 
$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$.
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. 
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