The Bertlmann-Martin inequalities (BMI) are studied in the three-body case. We consider distinguishable particles and scalar interactions. The systems are described by Schrödinger equations with local potentials. Under these conditions we show that the lower bound character of the BMI is preserved. We discuss the question of the correction factors transforming the inequalities into approximate or exact relationships. As illustrative example, a simple model in the (D = 1)-dimensional space is considered, for which the exact solution is known. We also study what can be learned from the Hartree approximation and the hyperspherical method in the central potential approximation with respect to the BMI inequalities and correction factors.

We propose a new integrable Hamiltonian describing two interacting particles in a harmonic mean field in D = 1 dimensional space. This model is found to be both supersymmetric and shape invariant. We show that for a given domain of the coupling constant the irregular solution is acceptable and contributes to the spectrum. We also discuss two inequalities of the Bertlmann–Martin type, which link the ground state mean-square radius to the lowest excitation energy.

Following Bertlmann and Martin, we derive an ensemble of recurrent inequalities in the framework of the Schrödinger equation in the 2-dimensional space. They link the moments of the ground state density to the energy differences between the ground state and the lowest state of each eigenvalue of the 2-dimensional angular momentum operator. Their application requires local potentials having azimuthal symmetry. We discuss the possibility of their extension beyond azimuthal symmetry by means of the Calogero model.

Starting from the minoring procedure of sum rules proposed by Bertlmann and Martin, we establish relationships for the average values of x2 and x4 for the ground and first excited states in D=1. These relations require only the knowledge of the energy spectrum. Thus, for a given potential, extension to its supersymmetric partner is immediate.

In the three-dimensional Schrodinger equation, the generalized Bertlmann-Martin inequalities connect the moments of the ground state density to the energy differences between the lowest level of each angular momentum e and the ground state. They are discussed in the case of the power-law potentials, as well as the In r potential. Use is made of the derived moments to reconstruct the form factor F(q), i.e., the Fourier transform of the ground state density. Pade approximants are used to describe the high q behavior of the form factor when only a limited number of low order moments are known. The estimate of the ground state density at the origin is also discussed. (C) 2003 Elsevier Inc. All rights reserved.