Let L(p)u = D4u - (p1u’)’ + p2u be a fourth-order differential operator acting on L2[0; 1] with p ≡ (p1; p2) belonging to L2ℝ[0, 1] x L2ℝ[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 L2ℝ[0, 1] x L2ℝ[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.
Copyright © 1998 SIAM. This article first appeared in SIAM Journal on Mathematical Analysis 29, no. 4 (July 1998): 935-66. doi:10.1137/s0036141096311198.
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Caudill, Lester F., Peter A. Perry, and Albert W. Schueller. "Isospectral Sets for Fourth-Order Ordinary Differential Operators." SIAM Journal on Mathematical Analysis 29, no. 4 (July 1998): 935-66. doi:10.1137/s0036141096311198.