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Few-Body Systems

The Bertlmann-Martin Inequalities in the Three-Body Problem

A. T. Kerris, F.-Z. Ighezou, R.J. Lombard, H. Ngô

Published Jan 25, 2006• 1 citation
Publisher
DOI
10.1007/s00601-005-0122-2

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The Bertlmann-Martin (BM) inequality for the rms radius is checked against the exact value in the case of a particle in a weak central potential having a single bound state. A number of usual potentials are considered in one and three dimensions. The BM inequality is poorly saturated but the correction factor needed to recover the exact rms radius is weakly depending on the shape of the potential, except for the Hulthén potential. For potentials having a hard-core component, we found that , which depends on the bound-state energy, tends to a universal curve as the hard core radius increases.
Reconstructing a radial (1D) quantum potential, [Formula: see text], from a few bound-state energies is a long-standing inverse problem because limited spectral data must constrain an entire potential. We present a Laplace-moment reconstruction pipeline that links the Bertlmann–Martin gap bound to generalized Bertlmann–Martin (GBM) even-moment ladders, continues the Laplace transform with Padé approximants, and inverts the transform to recover [Formula: see text] and [Formula: see text]. Odd moments are supplied by a physically consistent interpolation scheme. Benchmark settings and diagnostics for Coulomb, harmonic oscillator, Hulthén, Kratzer, and hyperbolic-well cases are stated so each approximation stage can be assessed under a common empirical basis. The conclusions are therefore limited to the reported benchmark settings rather than offered as universal method claims.