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

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 $M$.