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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
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