The present paper proposes a robust evaluation of any radial density at small distances using negative-order radial moments evaluated in momentum space. This evaluation provides a valuable insight into the behavior of a given radial density in the vicinity of $r=0$, and puts strong emphasis on the importance of measuring form factors at large squared four-momentum transfer, a domain essential for the determination of negative order moments. A specific attention is paid to the regularization scheme directly affecting the numerical determination of the radial density's parametrization. The proposed method is applied to non-relativistic study cases of the nucleon electric ($G_{En}, G_{Ep}$), and proton magnetic $G_{Mp}$ form factors. The validation is performed through comparison of the results of the approach to the analytically determined Maclaurin expansion - in the vicinity of $r=0$ - of the radial density function. The method is also applied to the relativistic Dirac form factor $F_1$ of the proton. In such a non-trivial case, the Maclaurin development might not exist for the radial density, rendering the determination from the proposed method extremely important.

A global analysis of proton electric form factor experimental data from Rosenbluth separation and low squared four-momentum transfer experiments is discussed for the evaluation of the spatial moments of the proton charge density based on the recently published integral method~\cite{Hob20}. Specific attention is paid to the evaluation of the systematic errors of the method, particularly the sensitivity to the choice of the mathematical expression of the form factor fitting function. Within this comprehensive analysis of proton electric form factor data, the moments of the proton charge density are determined for integer order moments, particularly: $\langle r^2 \rangle$=0.682(02)$_{Sta.}$(11)$_{Sys.}$~fm$^2$, $\langle r^3 \rangle$=0.797(10)$_{Sta.}$(58)$_{Sys.}$~fm$^3$, and $\langle r^4 \rangle$=1.02(05)$_{Sta.}$(31)$_{Sys.}$~fm$^4$.

The charge radius of the proton can be determined using two different kinds of experiments: the spectroscopy technique, measuring the hyperfine structure of hydrogen atoms, and the scattering technique, deducing the radius from elastic lepton scattering off a proton target. These two methods lead to quite different results, a discrepancy known as the "proton radius puzzle ". To shed light on this problem, we have proposed a novel method for the determination of spatial moments from densities expressed in the momentum space. This method provides a direct access not only to the second order moment, directly related to the proton radius, but to all moments of any real order larger than -3. The method is applied to the global analysis of proton electric form factor experimental data from Rosenbluth separation and low-$Q^2$ experiments, paying specific attention to the evaluation of the systematic errors. Within this analysis, the integer order moments of the proton charge density are evaluated, the moment of second order leading to a new determination of the proton charge radius.

For a particular case of three-body scattering in two dimensions, and matching analytical expressions at a transition point, we obtain accurate solutions for the hyperspherical adiabatic basis and potential. We find analytical expressions for the respective, asymptotic, inverse logarithmic and inverse power potential behaviours, that arise as functions of the radial coordinate. The model that we consider is that of two particles interacting with a repulsive step potential, a third particle acting as a spectator. The model is simple but gives insight, as the 2-body interaction is long ranged in hyperspherical coordinates. The fully interacting 3-body problem is known, numerically, to yield similar behaviours that we can now begin to understand. That, clearly, is the ultimate aim.

The precision of experimental data and analysis techniques is a key feature of any discovery attempt. A striking example is the proton radius puzzle where the accuracy of the spectroscopy of muonic atoms challenges traditional electron scattering measurements. The present work proposes a novel method for the determination of spatial moments from densities expressed in the momentum space. This method provides a direct access to even, odd, and more generally any real, negative and positive moment with order larger than −3. As an illustration, the application of this method to the electric form factor of the proton is discussed in detail.

We discuss the elastic scattering He-4+(24)wig at 50 MeV as a good example of a whole range of phenomena that compete in heavy ion reactions and can be identified in elastic scattering data. The reaction is dominated by deep optical potentials and shows all characteristics of a strongly refractive scattering: Fraunhofer diffractive oscillations at very forward angles, shallow Airy oscillation forward to a rainbow bump, significant increase of the cross section at large angles. Primitive evaluation of the deflection function indicates a strong orbiting singularity. The lowest order eikonal expansion of the scattering phase suggests a typical rainbow pattern, though the rainbow angle lies far beyond the physical angular range. The semiclassical uniform multireflection expansion of the scattering amplitude tells that the strong refractive effects associated with orbiting are carried by the internal barrier component which retains contributions from trajectories refracted by the spherical well. A complementary analysis in terms of unitary Regge pole amplitudes shows that the effects associated with the break in the exponential decay of the rainbow tail could be interpreted as simple diffraction due to pole(s) in the S-matrix. This strengthen the conjecture that orbiting and Regge poles are dual aspects of heavy ion scattering. The similarity with the Fresnel clothoide for light diffraction is exemplified with a schematic model that allows an almost exact calculation of the scattering amplitude.

We study the exact solutions of a particular class of N confined particles of equal mass, with \(N=3^k \ (k=2,3, \ldots ),\) in the \(D=1\) dimensional space. The particles are clustered in clusters of three particles. The interactions involve a confining mean field, two-body Calogero type of potentials inside the cluster, interactions between the centres of mass of the clusters and finally a non-translationally invariant N-body potential. The case of nine particles is exactly solved, in a first step, by providing the full eigensolutions and eigenenergies. Extending this procedure, the general case of N particles (\(N=3^k, \ k \ge 2\)) is studied in a second step. The exact solutions are obtained via appropriate coordinate transformations and separation of variables. The eigenwave functions and the corresponding energy spectrum are provided.

Relatively recently (A. Amaya-Tapia, S. Y. Larsen and M. Lassaut. Ann. Phys., vol. 306 (2011) 406), we presented a formula for the evaluation of the third Bose fugacity coefficient - leading to the third virial coefficient - in terms of three-body eigenphase shifts, for particles subject to repulsive forces. An analytical calculation for a 1-dim. model, for which the result is known, confirmed the validity of this approach. We now extend the formalism to particles with attractive forces, and therefore must allow for the possibility that the particles have bound states. We thus obtain a true generalization of the famous formula of Uhlenbeck and Beth (G.E. Uhlenbeck and E. Beth. Physica, vol. 3 (1936) 729; E. Beth and G.E. Uhlenbeck. ibid, vol.4 (1937) 915) (and of Gropper (L. Gropper. Phys. Rev. vol. 50 (1936) 963; ibid vol. 51 (1937) 1108)) for the second virial. We illustrate our formalism by a calculation, in an adiabatic approximation, of the third cluster in one dimension, using McGuire's model as in our previous paper, but with attractive forces. The inclusion of three-body bound states is trivial; taking into account states having asymptotically two particles bound, and one free, is not.

We review the semiclassical theory for heavy ion orbiting insisting on the connection with Regge poles and barrier-top resonances. Although the physical content of the phenomenon is well understood semiclassically, a clear signature is hard to be found because the relation between the observation angle and the deflection angle is not one to one.

We study the exact solutions of a particular class of N confined particles of equal mass, with N=3^k (k=2,3, … ), in the D=1 dimensional space. The particles are clustered in clusters of three particles. The interactions involve a confining mean field, two-body Calogero type of potentials inside the cluster, interactions between the centres of mass of the clusters and finally a non-translationally invariant N -body potential. The case of nine particles is exactly solved, in a first step, by providing the full eigensolutions and eigenenergies. Extending this procedure, the general case of N particles ( N=3^k, k ≥ 2 ) is studied in a second step. The exact solutions are obtained via appropriate coordinate transformations and separation of variables. The eigenwave functions and the corresponding energy spectrum are provided.

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.

We obtain an analytical expression for the energy eigenvalues of the Hyperbolical potential using an approximation of the centrifugal term. In order to obtain the /-states solutions, we use the the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space-time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the eigenfunctions are given in terms of hypergeometric functions. The present results are consistent with those obtained by others methods.

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.

We investigate the supersymmetry properties of energy dependent potentials in the D=1 dimensional space. We show the main aspects of supersymmetry to be preserved, namely the factorization of the Hamiltonian, the connections between eigenvalues and wave functions of the partner Hamiltonians. Two methods are proposed. The first one requires the extension of the usual rules via the concept of local equivalent potential. In this case, the superpotential becomes depending on the state. The second method, applicable when the potential depends linearly on the energy, is similar to what has been already achieved by means of the Darboux transform.

The � � interaction potential is obtained within the double folding model with density-dependent Gogny effective interactions as input. The one nucleon knock-on exchange kernel including recoil effects is localized using the Perey-Saxon prescription at zero energy. The Pauli forbidden states are removed thanks to successive supersymmetric transformations. Low energy experimental phase shifts, calculated from the variable phase approach, as well as the energy and width of the first 0 + resonance in 8 Be are reproduced with high accuracy.

We consider attractive power-law potentials depending on energy through their coupling constant. These potentials are proportional to 1/|x| m with m ≥ 1 in the D = 1 dimensional space, to 1/r m with m ≥ 2 in the D = 3 dimensional space. We study the ground state of such potentials. First, we show that all singular attractive potentials with an energy dependent coupling constant are bounded from below, contrarily to the usual case. In D = 1, a bound state of finite energy is found with a kind of universality for the eigenvalue and the eigenfunction, which become independent on m for m > 1. We prove the solution to be unique. A similar situation arises for D = 3 for m > 2, except that, in this case, the solution is not directly comparable to a bound state: the wave function, though square integrable, diverges at the origin.

The alpha - alpha interaction potential is obtained within the double folding model with density-dependent Gogny effective interactions as input. The one nucleon knock-on exchange kernel including recoil effects is localized using the Perey-Saxon prescription at zero energy. The Pauli forbidden states are removed thanks to successive supersymmetric transformations. Low energy experimental phase shifts, calculated from the variable phase approach, as well as the energy and width of the first 0(+) resonance in Be-8 are reproduced with high accuracy.

In one of the very few exact quantum mechanical calculations of fugacity coefficients, [L.R. Dodd, A.M. Gibbs. J. Math. Phys. 15 (1974) 41] obtained b(2) and b(3) for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For b(2), we have shown [A. Amaya-Tapia, S.Y. Larsen, M. Lassaut. Mol. Phys. 103 (2005) 1301-1306. < arXiv:physics/0405150 >] that Dodd and Gibbs' result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms, usually contributing only in one dimension. Now, we develop an exact expression for b(3) - b(3)(0) (where (0)b(3) is the free particle fugacity coefficient) in terms of sums and differences of three-body eigenphase shifts. Further, we show that if we obtain these eigenphase shifts in a Distorted-Born approximation, then, to first order, we reproduce the leading low temperature behaviour, obtained from an expansion of the twofold integral of Dodd and Gibbs. The contributions of the oscillating terms cancel.The formalism that we propose is not limited to one dimension, but seeks to provide a general method to obtain virial coefficients, fugacity coefficients, in terms of asymptotic quantities. The exact one dimensional results allow us to confirm the validity of our approach in this domain. (C) 2010 Elsevier Inc. All rights reserved.

Double folding alpha-alpha potentials based on the density dependent n-n Cogny interactions are constrained to reproduce the l=0 resonance in (8)Be. Shallow potentials are obtained by successive supersymmetric transformations which eliminate the l = 0 Pauli forbidden states. Implication for the EOS of cold alpha-matter are discussed.

Double folding α – α potentials based on the density dependent n – n Gogny interactions are constrained to reproduce the l = 0 resonance in 8Be. Shallow potentials are obtained by successive supersymmetric transformations which eliminate the l = 0 Pauli forbidden states. Implication for the EOS of cold α-matter are discussed.