**5. Concluding Remarks**

Using the instructive soluble example of electron scattering by the hard-sphere potential and delta-shell potential well, we for the first time explicitly obtained the angular time-delay <sup>Δ</sup>*t*(*k*, *θ*) in terms of the scattering phase shifts *<sup>δ</sup>l*(*k*) and their energy derivatives *<sup>δ</sup>l*(*k*). We demonstrated the complexity of <sup>Δ</sup>*t*(*k*, *θ*) as a function of the incoming electron energy *E* and the scattering angle *θ*. We saw that <sup>Δ</sup>*t*(*k*, *θ*) and the function Δ*t*(*k*)-, even averaged over proper intervals of *E* and *θ*, are more sensitive to the scattering phases than the absolute cross section *<sup>σ</sup>tot*(*k*) and even the differential in angle scattering cross section that is proportional to | *f*(*k*, *θ*)|2. This is because the time delay functions depend not only on the cross section phases, but also upon their energy derivatives. This makes theoretical and experimental investigation of time delays a promising direction of research in the area of atomic scattering.

**Author Contributions:** M.Y.A., A.S.B. and I.W. contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are available upon request.

**Acknowledgments:** In this section you can acknowledge any support The authors are thankful to Eric Morrow for proofreading the manuscript.

**Conflicts of Interest:** The authors declare no conflicts of interest.
