J. London Math. Soc. (2) 88, 801-828 (2013). [DOI: 10.1112/jlms/jdt041]

Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl

We discuss inverse spectral theory for singular differential operators on arbitrary intervals (a,b) ⊆ ℝ associated with rather general differential expressions of the type
τ f = 1/r ( - (p[f' + s f])' + s p[f' + s f] + qf),
where the coefficients p, q, r, s are Lebesgue measurable on (a,b) with p-1, q, r, s ∈ L1loc((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 Schrödinger operators with distributional potentials and Sturm-Liouville operators in impedance form are isolated, in particular.

MSC2010: Primary 34B24, 34L05; Secondary 34L40, 46E22.
Keywords: Sturm-Liouville operators, distributional coefficients, inverse spectral theory.

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