Article

**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
where the coefficients

*(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),*

*p*,*q*,*r*,*s*are Lebesgue measurable on*(a,b)*with*p*,^{-1}*q*,*r*,*s ∈ L*and real-valued with^{1}_{loc}((a,b); dx)*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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