Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems
Let \(E_ n\) be the eigenvalues of the Laplace-Beltrami operator on a \(d\)-dimensional compact smooth closed Riemannian manifold \(M\), \(-\Delta \varphi_ n= E_ n \varphi_ n\) and let \(N(E)= \#\{ E_ n\leq E\}\) be the spectral function of this operator. A classical problem is: What is the asymptotic behavior of the error term (in the Weyl formula) \[ D(E)= N(E)- {{\text{Vol } M\cdot \text{Vol } \Omega_ d} \over {(2\pi)^ d}} E^{d/2}, \] where \(E\to\infty\), where \(\Omega_ d= \{x\in \mathbb{R}^ d\), \(| x|\leq 1\}\). In the present work the authors study this problem in the case when \(M\) is a two-dimensional torus \(\mathbb{R}^ 2/ (2\pi a_ 1 \mathbb{Z}\oplus 2\pi a_ 2\mathbb{Z})\) and obtain interesting results, in particular if \({{a_ 1} \over {a_ 2}}\) is transcendental. The results are extended to a general class of \((C^ 7\)-smooth convex) ovals containing the origin with positive curvature such that the Fourier frequencies of a certain normalized error term satisfy some linear independence properties.
scientific article; zbMATH DE number 558979