Error Estimates for Non-selfadjoint Inverse Sturm-liouville Problems with Finite Spectral Data

This extended abstract is a summary of the main results in [4]. We consider a stability result for the inverse problem associated with the Sturm-Liouville equation −y′′ + q0(x)y = λy, x ∈ (0, 1), in which the potential q0 ∈ L(0, 1) is allowed to be complex-valued and the spectral data consists of the firstN Dirichlet-Dirichlet eigenvalues and the firstN Dirichlet-Neumann eigenvalues, determined to within an accuracy ε. As the spectral data is finite the problem may be expected to have infinitely many solutions (this appears to be unproven in the non-selfadjoint case). The usual philosophy in the numerical analysis literature is to construct recovery algorithms which select one of the infinitely many possible solutions. Numerical experiments are then carried out in which finite spectral data are generated from some known potential and the algorithm is declared to be good or bad according to how well it manages to recover the selected potential, in some norm. This process is meaningless unless one can prove that all of the infinitely many solutions to the finite data inverse problems are ‘close’, in some suitable sense. The point of this article is to establish such results. For reviews of reconstruction methods for inverse Sturm-Liouville problems see Rundell [10] and McLauglin [6]. There appears to be little published on stability for inverse Sturm-Liouville problems with finite data. For full data and a real potential, Ryabushko [12] proves the result