Average Error for Spectral Asymptotics on Surfaces

Abstract

Let $N(t)$ denote the eigenvalue counting funtion of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\tilde{N}(t)=At+B{t}^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac{1}{t}{\int }_0^{t}D(s)ds$ for $D(t)=N(t)-\tilde{N}(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.

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