An Inversion Inequality for Potentials in Quantum Mechanics

We suppose: (1) 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; and (2) that the potential shape can be expressed in the form f(x) = g(x^2), where g is monotone increasing and convex. The inversion inequality f(x) <= fbar(1/(4x^2)) is established, in which the `kinetic potential' fbar(s) is related to the energy function F(v) by the transformation: fbar(s) = F'(v), s = F(v) - vF'(v) As an example f is approximately reconstructed from the energy function F for the potential f(x) = x^2 + 1/(1+x^2).Comment: 7 pages (plain Tex), 2 figures (ps

Constructive inversion of energy trajectories in quantum mechanics

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.Comment: 16 pages in plain TeX with 5 ps figure

Functional inversion for potentials in quantum mechanics

Let E = F(v) be the ground-state eigenvalue of the Schroedinger Hamiltonian H = -Delta + vf(x), where the potential shape f(x) is symmetric and monotone increasing for x > 0, and the coupling parameter v is positive. If the 'kinetic potential' bar{f}(s) associated with f(x) is defined by the transformation: bar{f}(s) = F'(v), s = F(v)-vF'(v),then f can be reconstructed from F by the sequence: f^{[n+1]} = bar{f} o bar{f}^{[n]^{-1}} o 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.Comment: 14 pages, 2 figure