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 [B. Gao, Phys. Rev. A \textbf{58}, 4222 (1998)] is generalized to higher orders for angular momentum $l\ge 2$ 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 $l\ge 2$ 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 $l$.