On the regularity of the density of electronic wavefunctions

Electronic optics in graphene in the semiclassical approximation

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter $h \ll 1$. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intens…

$N$体Schrödinger演化方程的混合正则性与稀疏网格逼近

Mixed regularity and sparse grid approximations of $N$-body Schrödinger evolution equation

本文分析了含时$N$体电子系统的波函数混合正则性，据此开发了稀疏网格逼近（含GTO格式）以降低计算复杂度，并在氦原子和氢分子上验证了其相比全网格离散化的高效性。

On Factorization of Molecular Wavefunctions

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions \{$Φ$\} of the molecular Schrödinger equation as originally proposed by Hunter in the 1970s. The idea is to represent $Φ$ in the form $φχ$ where $χ$ is \textit{purely} a function of the nuclear coordinates, while $φ$ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an `exact' representation of $Φ$ can be ach…