2014/06/04 by Gayet, Damien, Welschinger, Jean-Yves
#FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1406.0934
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 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.