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In this work we fit neutron -12 C elastic scattering angular distributions in the energy range 12 to 20 MeV, by adding a velocity dependent term to the optical potential. This term introduces a wave function gradient, whose coefficient is real and position dependent, and which represents a nonlocality. We pay special attention to the prominent backscattering minima which depend sensitively on the incident energies, and which are a tell-tale of nonlocalities. Reasonable fits to the analyzing power data are also obtained as a by-product. All our potentials have the form of conventional Woods -Saxon shapes or their derivatives. The number of our parameters (12) is smaller than the number for other local optical potentials, and they vary monotically with energy, while the strengths of the real and imaginary parts of the central potential are nearly constants. Our nonlocality is in contrast to other forms of nonlocalities introduced previously.

PACS numberThe method is an extension to negative energies of a spectral integral equation method to solve the Schroedinger equation, developed previously for scattering applications. One important innovation is a re-scaling procedure in order to compensate for the exponential behaviour of the negative energy Green's function. Another is the need to find approximate energy eigenvalues, to serve as starting values for a subsequent iteration procedure. In order to illustrate the new method, the binding energy of the He-He dimer is calculated, using the He-He TTY potential. In view of the small value of the binding energy, the wave function has to be calculated out to a distance of 3000 a.u. Two hundred mesh points were sufficient to obtain an accuracy of three significant figures for the binding energy, and with 320 mesh points the accuracy increased to six significantfigures. An application to a potential with two wells, separated by a barrier, is also made.

Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are i) a recently developed spectral type integral equation method based on Chebyshev expansions, ii) a finite element expansion, and iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods i) and ii), achieving an accuracy of ten significant figures with a double precision calculation. Method iii) achieves seven significant figures. The scattering length and effective range are also obtained.1

The nonlocal Schrodinger equation is solved rigorously in a microscopic folding model, incorporating both direct and knock-on exchange potentials, for n-0 scattering at laboratory energies of 20 and 50 MeV. The model uses the complex and density dependent n-n interaction of N. Yamaguchi et al. , uses harmonic oscillator wave functions for the bound nucleons, and calculates the scattering wave function for this nonlocal problem using a Bessel-Sturmian expansion method incorporating correct boundary conditions. All spins are neglected. The local phase-equivalent potential is obtained from the scattering matrix elements at a given energy by using the iterative perturbative inversion method. This representation allows comparison between the microscopic model and a phenomenological potential, showing good agreement for the local real part of the potential at 20 MeV. From the ratio of the wave functions for the nonlocal potential and for the potential calculated by inversion, a Percy damping factor (PDF) is obtained which is of similar form to the well-known Percy-Buck prescription for the PDF for a Gaussian nonlocality of the conventional range of 0.85 fm. The signi6cance of these results for distorted wave Born approximation calculations is discussed.PACS number(s): 24.10.Ht, 25.40.Dn