Eigenfunctions and nodal sets

Eigenfunctions and nodal sets

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Steve Zelditch, Northwestern University
Fine Hall 314

Nodal sets are the zero sets of eigenfunctions of the Laplacian on a Riemannian manifold (M, g). If $\Delta \phi = \lambda^{2 \phi}$, then $\phi$ is somewhat analogous to a polynomial of degree $\lambda$ and its nodal set is somewhat analogous to a real algebraic variety of this degree. The analogy is closest if (M, g) is real analytic. Donnelly-Fefferman then showed that the hypersurface volume is of order $\lambda$. The physicists make the bold conjecture that, if the geodesic flow is chaotic, then the nodal sets become equidistributed with respect to the volume form. This is far beyond current technology, but I will show that if one complexifies everything--the manifold, the eigenfunctions and the nodal sets--then one can obtain equidistribution results in the ergodic case. The main new result is that intersections of the complex nodal set with a typical complex geodesic in the ergodic case becomes concentrated on the real geodesic in a uniform way with respect to arc-length. This is almost the physics conjecture.