H. Fiedeldey

Inverse problem for potential scattering at fixed energy

The inverse problem for quantal potential scattering at fixed energy is solved exactly for a scattering function which has the form of a product of a complex rational function of angular momentum times the scattering function of a given reference potential. Schematic numerical studies indicate the viability of the method in realistic applications.

A proposal for the determination of the phases in specular neutron reflection

Abstract A method is proposed for solving the phase problem in specular neutron reflection. A differential equation for the reflection phase is derived and solved in terms of the reflectivity and the dwell time. This opens the possibility of determining surface profiles from measured quantities by inversion. A schematic example is presented.

Nonlocal potentials and their exact and approximate local and velocity-dependent equivalents

We show that good approximations to the exact equivalent local potential (ELP) and damping factor of a nonlocal Perey-Buck potential can be calculated in the partial wave WKB approximation of Horiuchi. The exact ELP and damping factor are obtained by means of a method previously given by one of us. We also confirm that an approximate ELP proposed by Bauhoff et al. is of comparable accuracy as the Horiuchi approximation. Thesel-dependent ELPs exhibit reduced attraction in the interior and provide a test for higher order WKB approximations. We subsequently obtain an equivalent velocity dependent potential (EVDP) which is even exactly wave function equivalent to the original nonlocal potential. This almost local potential, unlike the trivial equivalent local potential, is smooth and well-behaved and is therefore particularly useful in nuclear reactions where the off-shell behaviour of the potential is important.

An optical potential from inversion of the 350 MeV 16O-16O scattering data

Abstract A quantal inversion of the 16O-16O scattering data at 350 MeV yields an optical potential which gives an excellent fit ( χ 2 F =1.65 ) to the measured cross-section. The real part of this potential is shallower than any potential used by others for distances between 2 and 6 fm. The imaginary potential is also relatively weak. This potential does not favour a rainbow interpretation of the structure in data observed at large scattering angles.

FURTHER INVESTIGATIONS OF OFF-SHELL EFFECTS IN THE TRITON.

Abstract We investigate the off-shell variation of the binding energy of the model triton E T for exactly phase-equivalent separable potentials of rank two. A physical explanation for the large off-shell effects previously found in E T is given which also explains the behaviour of these potentials in nuclear matter as reported by Srivastava et al . The arbitrary attractive form factor g ( k ) is with one exception chosen to be equal to a Yamaguchi form factor with strength and range parameters γ and β, and binding energy E B . With improved accuracy we can now show that E T ( γ , β ) reaches a lower limit of about 8.2 MeV if γ β 2 ⪢ 1 or E B ⪢ E D , the deuteron binding energy. This lower limit is shown to be independent of γ and only weakly dependent on β. The dependence on β is shown to be due to the far-off-shell behaviour of the T -matrix. In general we make it plausible that E T is mainly determined by the near-on-shell T -matrix for short-range potentials. It is also shown that for all potentials with E B significantly different from E D the UPA is a good approximation and the deuteron wave function Ψ D does not differ greatly. We also provide evidence that if Ψ D is given in addition to δ ( k ) and E d then the T -matrix is nearly completely determined even if E B ≈ E D . For those limiting potentials ( E B = E D ) which produce large values of E T up to 16.2 MeV, the UPA is shown to fail completely, due to a mismatch between δ ( k ) and Ψ D . This explains the large values of E T in those cases. All the limiting potentials are shown to be long-range potentials. It is also found that an approximately linear relation exists between E T (if E B ≠ E D ) and the zero-energy wound integral I 0 in analogy with the nuclear-matter results of Haftel and Tabakin.

Inverse problem for potential scattering at fixed energy. II.

The inverse problem for quantal potential scattering at fixed energy is solved for a class of scattering functions which represent nonrational modifications of the rational functions of angular momentum considered in the “Bargmann” method of previous work. The new scheme has a wider range of applicability. This is demonstrated by means of numerical examples.

Several versions of the integro-differential equation approach to bound systems

We find that a single two-variable integro-differential equation, which includes all two-body correlations, produces results for three- and four-body bound systems in good agreement with those obtained with the most accurate methods and also that for sixteen fermions, interacting by means of local Wigner-type potentials our results agree with those obtained with the Variational Monte-Carlo and the Fermi-Hypernetted chain methods.This equation includes a hypercentral potential. When it is set equal to zero it reduces to theS-state projected potential version of the formalism, which for three nucleons is identical to the exact Faddeev equation forS-state projected potentials.We show that the inclusion of the hypercentral component of the two-body local potential, which operates on all orbitals, takes the effect of the higher partial waves largely into account without the need of solving a system of coupled integro-differential equations. We, furthermore, show that by using the Weight-Function Approximation our integro-differential equation is transformed into a simple inhomogeneous differential equation, which becomes accurate forA≲16.The major advantages of our approximate method are, firstly, that unlike the exact Faddeev-Yakubovsky and Alt-Grassberger-Sandhas equations or the Monte-Carlo methods, it does not become rapidly unmanageable, but even simplifies with increasingA and, secondly, the speed and simplicity of our numerical calculations.

An integro-differential equation approach to the many-body problem

Abstract An integro-differential equation in two variables which includes two-body correlations, has been derived for a many-body system of identical particles. This equation has been obtained by means of a complete expansion in terms of the hyperspherical harmonic potential basis. It is proved that this equation is identical to the Faddeev equation for a system of three bosons. The implications of this result for many-body systems are discussed.

Quark-quark potentials from the inversion of baryon-spectra and its application to the Roper resonance

We show that it is in principle possible to determine the quark-quark potential, in the nonrelativistic potential model, from the baryon spectrum. The method we propose is based on the fact that the lowest order of the hyperspherical-harmonic expansion method, the hypercentral approximation, is an excellent approximation for confining quark-quark potentials. However, our method applies to all cases where this assumption is valid. Using standard inverse spectrum techniques adapted to our problem we invert the baryon spectrum to obtain the hypercentral potential in the hyperradius of the three-quark system. By means of a new exact relation based on the Abel integral equation, we can invert the hypercentral potential to determine the quark-quark potential.A first application of this new method to the inversion of thes-state baryon spectrum demonstrates in a model-independent way the inability of nonrelativistic two-quark potentials to reproduce the Roper resonance without violating the QCD-motivated concavity requirement.

Potential inversion for p- and α-scattering at fixed energy

Abstract The previous rational and nonrational Bargmann-type schemes for solving the inverse scattering problem are combined and generalized to include the long-range Coulomb force. The method is applied to p-58Ni, α-α and α-12C scattering with excellent results. Comparisons are made with the Newton-Sabatier inversion method as modified by Munchow and Scheid, and with the WKB-inversion method of Kujawski.