Average Error for Spectral Asymptotics on Surfaces

2013/01/21 by Robert S. Strichartz, Strichartz, Robert S.
Computer Science · Mathematics · #47A10 #58C40 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1301.4963

openalex publication_date 2013/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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) = (1)/(t) ∫₀^tD(s)ds for D(t) = N(t) - 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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