Approximating Maximally Localized Wannier Functions with Position Scaling-Eigenfunction

Position scaling-eigenfunctions are generated by transforming compactly supported orthonormal scaling functions and utilized for faster alternatives to maximally localized Wannier functions (MLWFs). The position scaling-eigenfunctions are first applied to numerical procedures solving Schr\"odinger and Maxwell's equations, and the solutions well agree with preceding results. Subsequently, by projecting the position scaling-eigenfunctions onto the space spanned by the Bloch functions, approximated MLWFs are obtained. They show good agreements with preceding results using MLWFs. In addition, analytical explanations of the agreements and an estimate of the error associated with the approximation are provided.