Analytic description of atomic interaction at ultracold temperatures: The case of a single channel

We present analytic descriptions of atomic interaction at ultracold temperatures using both single-channel and multichannel quantum-defect theories. In the case of a single channel, addressed in this paper, the expansion of Gao [Phys. Rev. A 58, 4222 (1998)] is generalized to higher orders for angular momentum l≥2 to give a more complete description of ultracold scattering, including an analytic description of ultracold shape resonances of arbitrary l. We also introduce a generalized scattering length that is well defined and useful for all partial waves to replace the traditional definition that fails for l≥2 due to the long-range van der Waals interaction. The results are used in a companion paper to derive analytic descriptions of atomic interaction around a magnetic Feshbach resonance of arbitrary angular momentum l.