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 $L + \rho$, where L is a fixed, second-order, selfadjoint, ${C^\infty }$ linear elliptic partial differential operator on a compact manifold (without boundary) and $\rho$ is a ${C^\infty }$ function. It is shown that, for almost all $\rho$, i.e. for a residual set, the eigenvalues of $L + \rho$ 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.