Functional inversion for potentials in quantum mechanics

Let E = F(v) be the ground-state eigenvalue of the Schrodinger Hamiltonian H = - Delta + vf(x), where the potential shape f(x) is symmetric and monoto ne increasing for x > 0, and the coupling parameter v is positive. if the k inetic potential (f) over bar(s) associated with f(x) is defined by the tra nsformation (f) over bar(s)= F(v), s = F(v)- vF'(v), then f can be reconstr ucted from F by the sequence f([n+1]) = (f) over bar . (f) over bar([n]-1) . f([n]) Convergence is proved for special classes of potential shape; for other test cases it is demonstrated numerically. The seed potential shape f ([0]) need not be 'close' to the limit f. (C) 2000 Published by Elsevier Sc ience B.V. All rights reserved.