Genericity of simple eigenvalues for elliptic PDE’s

Abstract

The spectrum of a selfadjoint, C' linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator. The purpose of this article is to provide a proof of the following theorem. Main theorem. Let M be a compact, connected C' manifold without boundary. Let L be a sel/adjoint, C' linear elliptic differential operator on M. The set Ep e Cw(M): all eigenvalues of L + p are simple} is residual in C'(M). The term elliptic operator used in this article will always mean the operator is selfadjoint, C' and linear. The main theorem can be summarized by saying that almost all elliptic operators obtained by varying the zeroth order part of a given operator have only simple eigenvalues. For terminology, see ? 1. The main theorem was first announced in [1] for second-order operators, and the proof is the one given in [2]; it uses perturbation theory. A proof using transversality theory, applicable to operators whose eigenfunctions satisfy the strong unique continuation property, has been obtained by K. Uhlenbeck [6]. The theorem is particularly useful in obtaining genericity results about the eigenfunctions (see [1] , [2], [6]). 1. Notation; outline of proof. The manifold M and the operator L will be fixed throughout as in the theorem. For standard terminology on elliptic operators, see [3], [4]. Received by the editors June 4, 1973 and, in revised form, January 23, 1974. AMS (MOS) subject classifications (1970). Primary 35P05, 35J30; Secondary 47A55, 47B25, 58G99.