Phase shift effective range expansion from supersymmetric quantum mechanics

Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the potential at infinity which allows us to introduce adequate long-distance transformations. The approach is shown to be effective in getting the correct phase shift effective range expansion. Applications are considered for the $^1P_1$ and $^1D_2$ partial waves of the neutron-proton scattering.Comment: 6 pages, 3 figures, Revtex4, version to be publised in Physical Review

Imaging of onset and progression of Alzheimer's disease with voxel-compression mapping of serial magnetic resonance images

Background Early diagnosis and monitoring of the progression of Alzheimer's disease is important for the development of therapeutic strategies. To detect the earliest structural brain changes, individuals need to be studied before symptom onset. We used an imaging technique known as voxel-compression mapping to localise progressive atrophy in patients with preclinical Alzheimer's disease.Methods Four symptom-free individuals from families with early-onset Alzheimer's disease with known autosomal dominant mutations underwent serial magnetic resonance imaging (MRI) over 5-8 years. All four became symptomatic during follow-up. 20 individuals with a clinical diagnosis of probable Alzheimer's disease and 20 control participants also underwent serial MR imaging. A non-linear fluid matching algorithm was applied to register repeat scans onto baseline imaging. Jacobian determinants were used to create the voxel-compression maps.Findings Progressive atrophy was revealed in presymptomatic individuals, with posterior cingulate and neocortical temporoparietal cortical losses, and medial temporal-lobe atrophy. In patients with known Alzheimer's disease, atrophy was widespread apart from in the primary motor and sensory cortices and cerebellum, reflecting the clinical phenomenology.Interpretation Voxel-compression maps confirmed early involvement of the medial temporal lobes, but also showed posterior cingulate and temporoparietal cortical losses at presymptomatic stage. This technique could be applied diagnostically and used to monitor the effects of therapeutic intervention

Phase equivalent chains of Darboux transformations in scattering theory

We propose a procedure based on phase equivalent chains of Darboux transformations to generate local potentials satisfying the radial Schrodinger equation and sharing the same scattering data. For potentials related by a chain of transformations, an analytic expression is derived for the Jost function. It is shown how the same system of S-matrix poles can be differently distributed between poles and zeros of a Jost function that corresponds to different potentials with equal phase shifts. The concept of shallow and deep phase equivalent potentials is analyzed in connection with distinct distributions of poles. It is shown that phase equivalent chains do not violate the Levinson theorem. The method is applied to derive a shallow and a family of deep phase equivalent potentials describing the S-1(0) partial wave of the nucleon-nucleon scattering