How rare are multiple eigenvalues?

Abstract

Let A(q) be a differentiable family of self-adjoint operators on a Hilbert space H, indexed by a parameter q that belongs to a separable Banach manifold X. Assume that the spectrum of each operator A(q) is discrete, of finite multiplicity, and with no finite accumulation points. We introduce a new concept of codimension in infinite-dimensional space and then prove that under an appropriate transversality condition, related to the strong Arnold hypothesis, the members of the family A(q) having multiple eigenvalues form a set of codimension at least 2. Using this, we show that a generic member of the family A(q) has a simple spectrum (i.e., no repeated eigenvalues) and that any two values q1 and q2 of the parameter can be connected by an analytic curve γ in X such that A(q) has a simple spectrum for all q in the interior of γ. We then apply these results in two cases of physical interest: to the Laplace operator with the domain as parameter and to the Schrödinger operator with a symmetric potential as parameter. © 1999 John Wiley & Sons, Inc.