Accurate Calculation of Off-Diagonal Green Functions on Anisotropic Hypercubic Lattices

We present a method for accurate evaluation of the Green function $G(\omega ,r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_d)$ at any real frequency $\omega$ and any lattice vector $(r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_d)$ for a $d$-dimensional hypercubic lattice that may have anisotropic couplings $({\mathrm{\Omega }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\Omega }}_d)$. In this method, we start with an integral representation of $G$, split the oscillatory integrand into combinations of Hankel functions, and deform the integration paths into the complex plane to obtain rapidly convergent integrals. We also discuss an alternative approach using the Levin collocation method. We report values of the Green function at selected frequencies on the branch cut and selected lattice vectors.