  We introduce and analyze free energy landscapes defined by associating to any
point inside the sphere a free energy calculated on a thin spherical band
around it, using many orthogonal replicas. This allows us to reinterpret,
rigorously prove and extend for general spherical models the main ideas of the
Thouless-Anderson-Palmer (TAP) approach originally introduced in the 70s for
the Sherrington-Kirkpatrick model. In particular, we establish a TAP
representation for the free energy, valid for any overlap value which can be
sampled as many times as we wish in an appropriate sense. We call such overlaps
multi-samplable. The correction to the Hamiltonian in the TAP representation
arises in our analysis as the free energy of a certain model on an overlap
dependent band. For the largest multi-samplable overlap it coincides with the
Onsager reaction term from physics. For smaller multi-samplable overlaps the
formula we obtain is new. We also derive the corresponding TAP equation for
critical points. We prove all the above without appealing to the celebrated
Parisi formula or the ultrametricity property.
  We prove that any overlap value in the support of the Parisi measure is
multi-samplable. For generic models, we further show that the set of
multi-samplable overlaps coincides with a certain set that arises in the
characterization for the Parisi measure by Talagrand. The ultrametric tree of
pure states can be embedded in the interior of the sphere in a natural way. For
this embedding, we show that the points on the tree uniformly maximize the free
energies we define. From this we conclude that the Hamiltonian at each point on
the tree is approximately maximal over the sphere of same radius, and that
points on the tree approximately solve the TAP equations for critical points.
