Betti numbers of random nodal sets of elliptic pseudo-differential operators

Damien Gayet, Jean-Yves Welschinger
2017 Asian Journal of Mathematics  
Given an elliptic self-adjoint pseudo-differential operator P bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold M equipped with some Lebesgue measure, we estimate from above, as L grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of P with eigenvalues below L. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse
more » ... on of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on M .
doi:10.4310/ajm.2017.v21.n5.a2 fatcat:kx673tptlzgtnmolqb6uar645u