AVERAGE ERROR FOR SPECTRAL ASYMPTOTICS ON SURFACES

摘要

Let N(t) denote the eigenvalue counting function of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula (N) over tilde (t) - At + Bt(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) = 1/t integral(t)(0) D(s) ds for D(t) = N(t) - (N) over tilde (t). We present a conjecture for the asymptotic behavior of A(t), and study some examples that support the conjecture.