An inversion inequality for potentials in quantum mechanics

We suppose: (1) that the ground-state eigenvalue E = F(upsilon) of the Schr odinger Hamiltonian H = -Delta + upsilon f(x) in one dimension is known for all values of the coupling upsilon > 0; and (2) that the potential shape c an be expressed in the form f(x) = g(x(2)), where g is monotone increasing and convex. The inversion inequality f(x) less than or equal to f (1/4x(2)) is established, in which the "kinetic potential'' (f) over bar(s) is relat ed to the energy function F(upsilon) by the transformation {(f) over bar(s) = F' (upsilon), s = F(upsilon) - upsilon F'(upsilon)}. As an example, f is approximately reconstructed from the energy function F for the potential f (x) = ax(2) + b/(c + x(2)). (C) 1999 American Institute of Physics. [S0022- 2488(99)01705-3].