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Nodal sets of sums of eigenfunctions. (English) Zbl 0946.35055
Christ, Michael (ed.) et al., Harmonic analysis and partial differential equations. Essays in honor of Alberto P. Calderón’s 75th birthday. Proceedings of a conference, University of Chicago, IL, USA, February 1996. Chicago, IL: The University of Chicago Press. Chicago Lectures in Mathematics. 223-239 (1999).
The general goal of the paper under review is to extend a result of H. Donnelly and C. Fefferman [Nodal sets of eigenfunctions on Riemannian manifold, Invent. Math. 93, No. 1, 161-183 (1988; Zbl 0659.58047)] on nodal sets of eigenfunctions of the Laplacian to a sum of eigenfunctions. Precisely, the authors show how to estimate the vanishing rate of such sums and the size of their zero sets. The main tools of their proofs are based on Carleman inequalities.
For the entire collection see [Zbl 0932.00088].

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J15 Second-order elliptic equations
58J50 Spectral problems; spectral geometry; scattering theory on manifolds