  Consider two inverse problems for Sturm-Liouville problems on the unit
interval. It means that there are two corresponding mappings $F, f$ from a
Hilbert space of potentials $H$ into their spectral data. They are called
isomorphic if $F$ is a composition of $f$ and some isomorphism $U$ of $H$ onto
itself. A isomorphic class is a collection of inverse problems isomorphic to
each other. We consider basic Sturm-Liouville problems on the unit interval and
on the circle and describe their isomorphic classes of inverse problems. For
example, we prove that the inverse problems for the case of Dirichlet and
Neumann boundary conditions are isomorphic. The proof is based on the
non-linear analysis.
