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

Abstract

In this paper, we consider the optimal recovery of potentials for a Sturm-Liouville problem $-y''+qy= \lambda y, y(0)=0=y(1)-hy'(1), 0<h<1, q\in L^1[0,1]$ with only one given eigenvalue. Denote by $\lambda _n(q)$ the $n-$th eigenvalue of this problem. For $\lambda \in \mathbb {R}$, denote by $\Omega _{n}(\lambda )=\left\{ q: q \in L^{1}[0,1], \lambda _{n}(q)=\right.$ $\left. \lambda \right\}$, $n \ge 1$ and $E_{n}(\lambda )=\inf \left\{ \Vert q\Vert : q \in \Omega _{n}(\lambda )\right\} .$ The optimal recovery of potential function in this paper refers to finding the infimum of the $L^1$-norm for potential function in the set $\Omega _{n}(\lambda ).$ We will obtain a formula for $E_{n}(\lambda )$ 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 $L^1[0, 1]$.