The generalized stationary scattering problems
In this paper, a generalized stationary scattering function is introduced and studied for both Schrödinger and Dirac radial equations. The Schrödinger radial operator is defined as \[ \mathcal{L}f = -\frac{\mathrm{d}^2f}{\mathrm{d}r^2}+\left[\frac{\ell(\ell+1)}{r^2}+\phi(r)+q(r)\right]f \] with the functions \(\phi\) and \(q\) satisfying \[ \int_a^\infty |\phi'(r)|\,\mathrm{d}r + \int_a^\infty |\phi^2(r)|\,\mathrm{d}r+\int_0^a |\phi(r)|r\,\mathrm{d}r <\infty\,, \quad 0 < a < \infty\,, \] \[ \int_0^a r |q(r)|\,\mathrm{d}r+\int_a^\infty |q(r)|\,\mathrm{d}r < \infty\,,\quad a>0\,, \] \[ \int_a^\infty |\phi(r)|\,\mathrm{d}r = \infty\,. \] These conditions are e.g. satisfied if we choose \(\phi\) to be the Coulomb potential. The author proves that the solution \(y_\ell (r,\lambda)\) of the eigenvalue equation \(\mathcal{L}f = \lambda f\), which for \(r\to 0\) behaves as \(y_\ell (r,\lambda) \sim N r^{\ell+1}\), has for \(r\to \infty\) the form \[ y_\ell (r,\lambda) \sim \frac{1}{2i} \left(\mathrm{exp\,}(ikr)V_0 (r,\lambda) \overline{S(\ell,\lambda)}-\mathrm{exp\,}(-ikr)V_0^{-1} (r,\lambda) S(\ell,\lambda)\right)\,, \] where the deviation factor \(V_0\) does not depend on \(\ell\) and has the form \[ V_0 (r, \lambda) = \mathrm{exp\,}\left(-\frac{i}{2k}\int_a^r \phi(u)\,\mathrm{d}u\right) \text{ with } k = \sqrt{\lambda}. \] Then the stationary scattering function is defined as \(S_{\mathrm{st}} = S(\ell,\lambda)/\overline{S(\ell,\lambda)}\). A similar result is obtained also for the radial Dirac equation. See also the paper [\textit{L. Sakhnovich}, Complex Anal. Oper. Theory 12, No. 3, 767--776 (2018; Zbl 1390.34238)].
generalized stationary scattering function
scientific article; zbMATH DE number 6860215