Analyticity and uniform stability of the inverse singular Sturm--Liouville spectral problem

We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with real-valued distributional potentials in the Sobolev class W^{s-1}_2(0,1), s\in[0,1].Comment: 25 page

Eigenvalue asymptotics for Sturm--Liouville operators with singular potentials

We derive eigenvalue asymptotics for Sturm--Liouville operators with singular complex-valued potentials from the space W^{\al-1}_{2}(0,1), \al\in[0,1], and Dirichlet or Neumann--Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra.Comment: Final version as appeared in JF

Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators

For a given self-adjoint operator $A$ with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~$B$ and discuss the inverse problem of reconstructing $B$ from its spectrum.Comment: 19 page

Spectra of rank-one perturbations of self-adjoint operators

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic multiplicityComment: 26 page