Critical potentials of the eigenvalue ratios of Schrödinger operators

✍ Scribed by Songbo Hou

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 300 KB
• Volume: 70
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.12.024

✦ Synopsis

In this paper, we consider an operator H defined on a compact smooth n-manifold M by H = -+ q acting on functions with Dirichlet or Neumann boundary conditions in the case ∂ M = ∅ and with eigenvalues λ 1 (q) < λ 2 (q) ≤ λ 3 (q) ≤ • • •. We investigate critical potentials of the ratio of two consecutive eigenvalues considered as functionals on the set of bounded potentials having a given mean value on M. We obtain necessary and sufficient conditions for a potential to be a critical point of such a functional.