Abstract: In this talk I will discuss recent joint results with T. Aktosun on the Schrödinger
equation on the half line with a real-valued, integrable potential having a finite first moment. It is shown
that the potential and the boundary conditions are uniquely determined by the data containing the discrete
eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that
boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This
result extends the celebrated two-spectra uniqueness theorem of Borg and Marchenko to the case where
there is also a continuous spectrum.
Furthermore, I will present recent joint results with M. Lassas and V. Kurylev in a multi-dimensional
analogue of the Borg-Levinson two spectra theorem. We consider multi-dimensional Schrödinger
operators defined in a compact set, with Robin boundary condition. We prove that the eigenvalues,
and the Gateaux derivative of the eigenvalues with respect to the parameter in the Robin boundary
condition, uniquely determine both the potential and the Robin parameter.