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The Leray measure of nodal sets for random eigenfunctions on the torus. (English) Zbl 1153.35058
For \(E>0\), let \(N(E)\) be the dimension of the eigenspace of the Laplacian on the \(d\)-dimensional torus, i.e. \(N(E)\) is the number of the set of all \(\lambda\in \mathbb Z^d\) such that \(| \lambda| ^2 =E.\) Let \(b_\lambda, c_\lambda\) be independent standard Gaussian random variables such that \(b_{-\lambda}=b_\lambda\) and \(c_{-\lambda} =-c_\lambda\), and let \(L(f)\) be the Leray measure of the nodal set of the random eigenfunction
\[ f(x):= (2N(E))^{-1/2} \sum_{\lambda\in \mathbb Z^d: | \lambda | ^2=E} b_\lambda \cos 2\pi I \lambda\cdot x -c_\lambda \sin 2 \pi i \lambda\cdot x. \] The introduced Leray measure has the property that \(\mathbb E L(f)= (2\pi)^{-1/2}\) and \(\text{Var}(L(f))\) behaves as \((4\pi N(E))^{-1} \) as \(N\to \infty\).

35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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