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We propose a numerical method using the discrete variable representation(DVR) for constructing real-valued Wannier functions localized in a unit cellfor both symmetric and asymmetric periodic potentials. We apply these resultsto finding Wannier functions for ultracold atoms trapped in laser-generatedoptical lattices. Following Kivelson \cite{kivelson_wannier_1982}, for asymmetric lattice with inversion symmetry, we construct Wannier functions aseigen states of the position operators $\hat x$, $\hat y$ and $\hat z$restricted to single-particle Bloch functions belonging to one or more bands.To ensure that the Wannier functions are real-valued, we numerically obtain theband structure and real-valued eigen states using a uniform Fourier grid DVR.We then show by a comparison of tunneling energies, that the Wannier functionsare accurate for both inversion symmetric and asymmetric potentials to betterthan ten significant digits when using double-precision arithmetic. Thecalculations are performed for an optical lattice with double-wells per unitcell with tunable asymmetry along the $x$ axis and a single sinusoidalpotential along the perpendicular directions. Localized functions at the twopotential minima within each unit cell are similarly constructed, but using asuperposition of single-particle solutions from the two lowest bands. Wefinally use these localized basis functions to determine the two-bodyinteraction energies in the Bose-Hubbard (BH) model, and show the dependence ofthese energies on lattice asymmetry.

Wannier functions using a discrete variable representation for optical lattices