Multichannel quantum defect theory with numerical reference functions: Applications to cold atomic collisions

We develop a method for the calculation of multichannel wavefunctions in the spirit of quantum defect theory using numerically calculated reference functions. We first verify our method by calculating cold collisional properties of $^{85}$Rb and $^6$Li in the presence of external magnetic fields tuned across specific $s$-wave Feshbach resonances and thereby reproducing known results. We then calculate recently discovered $d$-wave Feshbach resonance [Phys. Rev. Lett. {\bf 119}, 203402 (2017)] in $^{87}$Rb-$^{85}$Rb cold collisions by our method. Our numerical results on this $d$-wave resonance agree reasonably well with the experimental ones. Our method is applicable to any arbitrary form of potentials and any arbitrary range of energies around threshold. The implementation of our method to any multichannel two-body scattering problem is straightforward.