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Register a new userThe effective interaction hyperspherical harmonics method for non-local potentials
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A different formulation of the effective interaction hyperspherical harmonics (EIHH) method, suitable for non-local potentials, is presented. The EIHH method for local interactions is first shortly reviewed to point out the problems of an extension to non-local potentials. A viable solution is proposed and, as an application, results on the ground-state properties of 4- and 6-nucleon systems are presented. One finds a substantial acceleration in the convergence rate of the hyperspherical harmonics series. Perspectives for an application to scattering cross sections, via the Lorentz transform method are discussed.
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The Schroedinger equation is solved for an A-nucleon system using an expansion of the wave function in nonsymmetrized hyperspherical harmonics. Our approach is both an extension and a modification of the formalism developed by Gattobigio et al.. The extension consists in the inclusion of spin and isospin degrees of freedom such that a calculation with more realistic NN potential models becomes possible, whereas the modification allows a much simpler determination of the fermionic ground state. The approach is applied to four- and six-body nuclei (4He, 6Li) with various NN potential models. It is shown that the results for ground-state energy and radius agree well with those from the literature.
The non-symmetrized hyperspherical harmonics method for a three-body system, composed by two particles having equal masses, but different from the mass of the third particle, is reviewed and applied to the $^3$H, $^3$He nuclei and $^3_{Lambda}$H hyper-nucleus, seen respectively as $nnp$, $ppn$ and $NNLambda$ three-body systems. The convergence of the method is first tested in order to estimate its accuracy. Then, the difference of binding energy between $^3$H and $^3$He due to the difference of the proton and the neutron masses is studied using several central spin-independent and spin-dependent potentials. Finally, the $^3_{Lambda}$H hypernucleus binding energy is calculated using different $NN$ and $Lambda N$ potential models. The results have been compared with those present in the literature, finding a very nice agreement.
The method of effective interaction, traditionally used in the framework of an harmonic oscillator basis, is applied to the hyperspherical formalism of few-body nuclei (A=3-6). The separation of the hyperradial part leads to a state dependent effective potential. Undesirable features of the harmonic oscillator approach associated with the introduction of a spurious confining potential are avoided. It is shown that with the present method one obtains an enormous improvement of the convergence of the hyperspherical harmonics series in calculating ground state properties, excitation energies and transitions to continuum states.
The Hyperspherical Harmonics basis, without a previous symmetrization step, is used to calculate binding energies of the nuclear A=6 systems using a version of the Volkov potential acting only on s-wave. The aim of this work is to illustrate the use of the nonsymmetrized basis to deal with permutational-symmetry-breaking term in the Hamiltonian, in the present case the Coulomb interaction.
The effective-interaction theory has been one of the useful and practical methods for solving nuclear many-body problems based on the shell model. Various approaches have been proposed which are constructed in terms of the so-called $widehat{Q}$ box and its energy derivatives introduced by Kuo {it et al}. In order to find out a method of calculating them we make decomposition of a full Hilbert space into subspaces (the Krylov subspaces) and transform a Hamiltonian to a block-tridiagonal form. This transformation brings about much simplification of the calculation of the $widehat{Q}$ box. In the previous work a recursion method has been derived for calculating the $widehat{Q}$ box analytically on the basis of such transformation of the Hamiltonian. In the present study, by extending the recursion method for the $widehat{Q}$ box, we derive another recursion relation to calculate the derivatives of the $widehat{Q}$ box of arbitrary order. With the $widehat{Q}$ box and its derivatives thus determined we apply them to the calculation of the $E$-independent effective interaction given in the so-called Lee-Suzuki (LS) method for a system with a degenerate unperturbed energy. We show that the recursion method can also be applied to the generalized LS scheme for a system with non-degenerate unperturbed energies. If the Hilbert space is taken to be sufficiently large, the theory provides an exact way of calculating the $widehat{Q}$ box and its derivatives. This approach enables us to perform recursive calculations for the effective interaction to arbitrary order for both systems with degenerate and non-degenerate unperturbed energies.
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