ALBERT

Notes: An implementation of exterior complex scaling using the finite elements method with high degree polynomials is presented. We apply the method to find the resonances of the potential 7.5r 2e−r and of a phenomenological coupled channel model of the CaH molecule. In both cases the method is quickly convergent and extremely stable numerically. Convergence could be pushed to the point where the real parts of most resonance energies were independent of the complex scaling angle and of the exterior scaling radius within machine precision (14 significant digits). All imaginary parts were stable to at least eight significant digits. Several resonances of CaH which had evaded searches with a finite difference method could be located.

Notes: The radial equation (or set of equations) derived in scattering theory is analyzed by means of Titchmarsh–Weyl theory for singular second-order differential equations. In particular we have focused on the spectral density concept and the corresponding relation to the scattering cross section. The method of complex deformations is brought in as a necessary ingredient in the evaluation of the underlying pole strings, which together with the background build up the actual dispersion relation data. The analysis is supported by numerical applications to a centrifugal family of simple potentials.

Notes: The Titchmarsh–Weyl theory is applied to the Schrödinger equation in the case when the asymptotic form of the solution is not known. It is assumed that the potential belongs to the Weyl's limit-point classification. A rigorous analytical continuation of the Green's function, obtained from the solution regular at the origin and the square integrable Weyl's solution (regular at infinity), to the "unphysical'' Riemann energy sheet is carried out. It is demonstrated how the Green's function can be uniquely constructed from the Titchmarsh–Weyl m-function and its Nevanlinna representation. The behavior of the m-function in the neighborhood of poles is investigated. The m-function is decomposed in a, so called, generalized real part (Reg) and a generalized imaginary part (Img). Reg(m) is found to have a significant argument change upon pole passages. Img(m) is found to be a generalized spectral density. From the generalized spectral density, a spectral resolution of the differential operator and its resolvent is derived. In the expansion contributions are obtained from bound states, resonance states (Gamow states), and the "deformed continuum'' given by the generalized spectral density.The present expansion theorem is applicable to the general partial differential operator via a decomposition into partial waves. The numerical procedure involves all quantum numbers l and m, but for convenience, and with the inverse problem in mind, this study is focused on the case when the rotational quantum number equals zero. The theory is tested numerically and analyzed for an analytic model potential exhibiting a barrier and decreasing exponentially at infinity. The potential is Weyl's limit point at infinity and allows for an analytical continuation into a sector in the complex plane. An attractive feature of the generalized spectral density of the present potential is that the poles close to the real axis seem to exhaust or deflate the above-mentioned density inside the pole string. Outside this string the density rapidly approaches that of a free particle. This information is used to derive an approximate representation of the m-function in terms of poles and residues as well as free-particle background. In order to display the features mentioned above, the present study is accompanied with several plots of analytically continued quantities related to the Green's function.