## Stationary Wigner Equation with Inflow Boundary Conditions: Will a Symmetric Potential Yield a Symmetric Solution?

科学计算团队系列报告

报告人：卢朓 (北京大学)

报告时间：2018年10月26日下午2:30-3:30

报告地点：理科楼A404

邀请人：黄忠亿、吴昊

摘要：

It is highly attractive to model open quantum devices based on the Wigner function formalism. For the stationary Wigner transport equation, the well-posedness with inflow boundary condition(BC) is a long-standing open problem. Hence the applicability of the Wigner function formalism to quantum devices is lack of theoretic foundation.

In 2006, F. Rossi, et al, realized that a symmetric potential may produce a symmetric solution, even if the inflow BC is NOT symmetric. They argued this counter-intuitive conjecture by a formal analysis based on a Neumann series approach at first, and then illustrated that though an upwinding scheme always gives solutions without symmetry, a symmetry solution can be obtained by a central finite difference scheme.

Though one can deny directly their analysis by a counter example, their solid numerical evidence seems to push the Wigner function formalism to the wall: either the Wigner transport equation with inflow BC produce a non-physical solution, or its solution is completely unstable to perturbation that different numerical schemes give different solutions. It was pointed out therein that ``the Wigner transport equation may produce highly non-physical results'' and was pronounced in their paper title a ``failure of conventional boundary condition schemes''.

To straighten out such a confused situation, I will show in this talk that:(1) We proved Rossi's symmetry conjecture with a periodic potential. Hence the solution of stationary Wigner transport equation with inflow BC is always symmetric only if the potential is symmetric and periodic. Our proof is based on the well-posedness by Arnold, et al, without any additional prerequisite conditions.

(2) We carried out a thorough numerical investigation to show that different numerical schemes always produce symmetric solutions. This indicates us that the Wigner function formalism gives us a stable model. With our study, one may get back the confidence to the Wigner function formalism for quantum devices modelling and simulation.

报告人简介：

卢朓，2004年获北京大学计算数学专业博士学位(导师应隆安、张平文)。北京大学数学学院副教授、博士生导师，北京计算数学学会常务理事。能源与环境大数据研究中心研究员。北京大学应用物理与技术研究中心研究员。从事色散媒质的高阶数值方法，纳米半导体器件的建模、分析和数值模拟，量子动理学方程的数值方法和分析等方面的研究工作。已经发表SCI论文二十多篇。2003年获得中国计算数学学会青年优秀论文一等奖。2017年获得北京大学教学优秀奖（研究生部分）。