  We suppose that the ground-state eigenvalue E = F(v) of the Schroedinger
Hamiltonian H = -\Delta + vf(x) in one dimension is known for all values of the
coupling v > 0. The potential shape f(x) is assumed to be symmetric, bounded
below, and monotone increasing for x > 0. A fast algorithm is devised which
allows the potential shape f(x) to be reconstructed from the energy trajectory
F(v). Three examples are discussed in detail: a shifted power-potential, the
exponential potential, and the sech-squared potential are each reconstructed
from their known exact energy trajectories.
