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 ≡ (p₁,p₂) belonging to L²_ℝ[0, 1] × L²_ℝ[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 L²_ℝ[0, 1] × L²_ℝ[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.