Abstract. The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space W −1 2 (0, 1). The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

The spectral properties of Sturm-Liouville operators with an impedance in W 1 p (0, 1), for some p ∈ [1, ∞), are studied. In particular, a complete solution of the inverse spectral problem is provided.

We study an inverse spectral problem for a compound oscillating system consisting of a singular string and N masses joined by springs. The operator A corresponding to this system acts in L 2 (0, 1) × C N and is composed of a Sturm-Liouville operator in L 2 (0, 1) with a distributional potential and a Jacobi matrix in C N , which are coupled in a special way. We solve the inverse spectral problem for the operator A and describe explicitly the set of spectral data. We also exhibit a connection to related Sturm-Liouville problems with boundary conditions depending rationally on the spectral parameter.

We study the inverse spectral problem for a class of Bessel operators given in L 2 (0, 1) by the differential expressionwith κ ∈ N and a real-valued function v ∈ L p (0, 1), p ∈ [1, ∞), subject to various boundary conditions. We describe completely the spectral data of these operators, i.e., the spectra and corresponding norming constants, and give the algorithm of reconstruction of v from the spectral data.