Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

• Yifei Jia ^[1] ; Jiangang Qi ^[1] ; Jing Li ^[1]
  1. [1] Shandong University

     Shandong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 2, 2023
• Idioma: inglés
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• Resumen

  • In this paper, we consider the optimal recovery of potentials for a Sturm-Liouville problem −y + qy = λy, y(0) = 0 = y(1) − hy (1), 0 < h < 1, q ∈ L1[0, 1] with only one given eigenvalue. Denote by λn(q) the n−th eigenvalue of this problem.

    For λ ∈ R, denote by n(λ) = q : q ∈ L1[0, 1], λn(q) = λ}, n ≥ 1 and En(λ) = inf {q : q ∈ n(λ)}. The optimal recovery of potential function in this paper refers to finding the infimum of the L1-norm for potential function in the set n(λ). We will obtain a formula for En(λ) and specify where the infimum can be attained. Our results are closely related to the discontinuity of the eigenvalues with respect to the boundary conditions. Since the optimal recovery problem with only one fixed eigenvalue is just the duality problem to the extremum problem of eigenvalues, we also give the extremum of the n-th eigenvalue of a problem for potentials on a sphere in L1[0, 1].