Generic properties of eigenfunctions of elliptic partial differential operators

Abstract

The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form L + ρ L + \rho , where L is a fixed, second-order, selfadjoint, C ∞ {C^\infty } linear elliptic partial differential operator on a compact manifold (without boundary) and ρ \rho is a C ∞ {C^\infty } function. It is shown that, for almost all ρ \rho , i.e. for a residual set, the eigenvalues of L + ρ L + \rho are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.