dorsal/arxiv
View SchemaAn Inversion Inequality for Potentials in Quantum Mechanics
| Authors | Richard L. Hall |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9809019 |
| URL | https://arxiv.org/abs/quant-ph/9809019 |
| DOI | 10.1063/1.532862 |
| Journal | J.Math.Phys. 40 (1999) 2254-2258 |
Abstract
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).
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"abstract": "We suppose: (1) that the ground-state eigenvalue E = F(v) of the Schroedinger\nHamiltonian H = -Delta + vf(x) in one dimension is known for all values of the\ncoupling v \u003e 0; and (2) that the potential shape can be expressed in the form\nf(x) = g(x^2), where g is monotone increasing and convex. The inversion\ninequality f(x) \u003c= fbar(1/(4x^2)) is established, in which the `kinetic\npotential\u0027 fbar(s) is related to the energy function F(v) by the\ntransformation: fbar(s) = F\u0027(v), s = F(v) - vF\u0027(v) As an example f is\napproximately reconstructed from the energy function F for the potential f(x) =\nx^2 + 1/(1+x^2).",
"arxiv_id": "quant-ph/9809019",
"authors": [
"Richard L. Hall"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1063/1.532862",
"journal_ref": "J.Math.Phys. 40 (1999) 2254-2258",
"title": "An Inversion Inequality for Potentials in Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9809019"
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