Average error for spectral asymptotics  on surfaces

Robert S. Strichartz

doi:10.3934/cpaa.2016.15.9

Article Contents

2016, Volume 15, Issue 1: 9-39. Doi: 10.3934/cpaa.2016.15.9

This issue Previous Article Average number of lattice points in a disk Next Article Large time behavior of solutions for a nonlinear damped wave equation

Average error for spectral asymptotics on surfaces

Robert S. Strichartz^1,

1.
Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853

Received: June 2015

Revised: October 2015

Published: January 2016

Abstract / Introduction Related Papers Cited by

Abstract

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 $\widetilde N(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac 1 t \int^t_0 D(s)\,ds$ for $D(t)=N(t)-\widetilde N(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.

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Mathematics Subject Classification: 47A10, 58C40, 58J50.

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