On the moments of the ground and first excited states

PTHStarting 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

Bertlmann–Martin inequalities in D=2 and the Calogero model

PTHFollowing 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

A new model of the Calogero type

PTHWe 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