Equivalence of time-domain inverse problems and boundary spectral problems

We consider inverse problems for wave, heat and Schr"odinger-type operators and corresponding spectral problems on domains of ${\mathbf{R}}^n$ and compact manifolds. Also, we study inverse problems where coefficients of partial differential operator have to be found when one knows how much energy it is required to force the solution to have given boundary values, i.e., one knows how much energy is needed to make given measurements. The main result of the paper is to show that all these problems are shown to be equivalent.