Isospectral Sets for Fourth-Order Ordinary Differential Operators

Let L(p)u = D4u - (p1u')' + p2u be a fourth-order differential operator acting on L2[0,1] with $p\equiv (p1,p2) belonging to $L^2_\bbR[0,1]\times L^2_\bbR[0,1]$ and boundary conditions u(0)=u'(0)=u(1)=u'(1)=0. We study the isospectral set of L(p) when L(p) has simple spectrum. In particular we show that for such p, the isospectral manifold is a real-analytic submanifold of $L^2_\bbR[0,1]\times L^2_\bbR[0,1]$ which has infinite dimension and codimension. A crucial step in the proof is to show that the gradients of the eigenvalues of L(p) with respect to p are linearly independent: we study them as solutions of a non-self-ajdoint fifth-order system, the Borg system, among whose eigenvectors are the gradients.