Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions

Egor E. Chitorkin; Natalia P. Bondarenko

Home/Archive/arXiv

arXiv · 2305.01231

Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions

Egor E. Chitorkin·Natalia P. Bondarenko

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we for the first time get constructive solution for the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction.

Explore related subjects

math.SP

Keep this discovery

BibTeX RIS

Egor E. Chitorkin, Natalia P. Bondarenko. 2023-05-02. Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions. https://arxiv.org/abs/2305.01231

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗

arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗

arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators $P+V$, where $P$ is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and $V$ is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for $P$ imply $L^p$-$L^{p'}$ resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗