Constructive inversion of energy trajectories in quantum mechanics

We suppose that the ground-state eigenvalue E=F(upsilon) of the Schrodinger Hamiltonian H=-Delta+upsilon f(x) in one dimension is known for all values of the coupling upsilon>0. The potential shape f(x) is assumed to be symme tric, bounded below, and monotone increasing for x>0. A fast algorithm is d evised which allows the potential shape f(x) to be reconstructed from the e nergy trajectory F(upsilon). Three examples are discussed in detail: a shif ted power-potential, the exponential potential, and the sech-squared potent ial are each reconstructed from their known exact energy trajectories. (C) 1999 American Institute of Physics. [S0022-2488(99)01202-5].