Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

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 \[τf = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), \] where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s\in L_{\text{loc}}^1((a,b);dx)$ and real-valued with $p\ne 0$ 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 Schrödinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.

29 pages

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