Generalized Green’s functions and spectral densities in the complex energy plane

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Abstract
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.