Isospectral sets for fourth-order ordinary differential operators

Lester F. Caudill, Peter A. Perry, Albert W. Schueller

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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[0, 1] × L2[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[0, 1] × L2[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.

Original languageEnglish
Pages (from-to)935-966
Number of pages32
JournalSIAM Journal on Mathematical Analysis
Issue number4
StatePublished - Jul 1998


  • Inverse spectral problem
  • Ordinary differential equations

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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