Volume des ensembles nodaux des fonctions propres du Laplacien. (Volume of the nodal sets of the eigenvalues of the Laplacian). (French) Zbl 0589.58033

Bérard, Pierre

Sémin. Bony-Sjöstrand-Meyer, Équations Dériv. Partielles 1984-1985, Exp. No. 14, 9 p. (1985).

Let M be a compact Riemannian manifold. Suppose that \(\phi_ j\) are the eigenfunctions, with eigenvalue \(\lambda_ j\), of the Laplacian for M. Let \(V_ a(k)\) be the volume of the zero set of \(\sum^{k}_{j=1}a_ j\phi_ j\), for \(a\in RP^ k\). The author gives an asymptotic formula for the average value of \(V_ a(k)\), as \(k\to \infty\).

Reviewer: H.Donnelly

MSC:

58J50

Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

nodal set; eigenfunctions; Laplacian