We study new solvable few body problems consisting of generalizations of the Calogero and the Calogero–Marchioro–Wolfes three-body problems, by introducing non-translationally invariant three-body potentials. After separating the radial and angular variables by appropriate coordinate transformations, we provide eigensolutions of the Schrödinger equation with the corresponding energy spectrum. We found a domain of the coupling constant for which the irregular solutions are square integrable.
We propose and solve exactly the Schrödinger equation of a bound quantum system consisting in four particles moving on a real line with both translationally invariant four particles interactions of Wolfes type [1] and additional non translationally invariant four-body potentials. We also generalize and solve exactly this problem in any D-dimensional space by providing full eigensolutions and the corresponding energy spectrum. We discuss the domain of the coupling constant where the irregular solutions becomes physically acceptable PACS: 02.30.Hq, 03.65.-w, 03.65.GeThe paper is organized as follows. In section 2 we expose and solve the problem for the linear case. The section 3 is devoted to extension to D-dimensional problem. Our conclusions are drawn in section 4.
This work is devoted to the study of some exactly solvable quantum problems of four, five and six bodies moving on the line. We solve completely the corresponding stationary Schrödinger equation for these systems confined in an harmonic trap, and interacting pairwise, in clusters of two and three particles, by two-body inverse square Calogero potential. Both translationaly and non-translationaly invariant multi-body potentials are added. In each case, the full solutions are provided, namely the normalized regular eigensolutions and the eigenenergies spectrum. The irregular solutions are also studied. We discuss the domains of coupling constants for which these irregular solutions are square integrable. The case of a "Coulomb-type" confinement is investigated only for the bound states of the four-body systems.
In a previous paper, we have proposed a new integrable Hamiltonian describing two interacting particles in a harmonic mean field in D = 1 dimensional space. Here, we generalize this Hamiltonian to the D 2 dimensional space. We show that the system is exactly solvable for a certain domain of the interaction constant, and that the size of this domain increases with D. We also show this D-dimensional Hamiltonian to be supersymmetric and shape invariant, very much as in the D = 1 case.
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