Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

TL;DR

This paper develops inverse spectral theory for a broad class of singular Sturm-Liouville operators with distributional potentials, providing new results for spectral measure, two- and three-spectra, and local inverse problems.

Contribution

It introduces inverse spectral results for general differential expressions with distributional coefficients, extending classical theory to more singular cases.

Findings

Derived inverse spectral results from spectral measure, two-spectra, and three-spectra.

Included special cases like Schrödinger operators with distributional potentials.

Established local Borg-Marchenko-type inverse spectral results.

Abstract

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[\tau f = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), \] where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schr\"odinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.