19 research outputs found
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
Hamiltonian Transformation for Band Structure Calculations
First-principles electronic band structure calculations are essential for
understanding periodic systems in condensed matter physics and materials
science. We propose an accurate and parameter-free method, called Hamiltonian
transformation (HT), to calculate band structures in both density functional
theory (DFT) and post-DFT calculations with plane-wave basis sets. The cost of
HT is independent of the choice of the density functional and scales as
, where and are the number of
electrons and the number of -points. Compared to the widely used
Wannier interpolation (WI), HT adopts an eigenvalue transformation to construct
a spatial localized representation of the spectrally truncated Hamiltonian. HT
also uses a non-iterative algorithm to change the basis sets to circumvent the
construction of the maximally localized Wannier functions. As a result, HT can
significantly outperform WI in terms of the accuracy of the band structure
calculation. We also find that the eigenvalue transformation can be of
independent interest, and can be used to improve the accuracy of the WI for
systems with entangled bands.Comment: 5 pages, 4 figure