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


Ruo Li,Department of Mathematics,Peking University


2019.03.12 14:00-15:00


601, Pao Yue-Kong Library


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 opened 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 prounced 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 solution (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.


Ruo Li is now the distinguished professor Chang Jiang scholars in computational mathematics in School of Mathematical Sciences, Peking University. He got his PhD in Peking Univerisity in 2001. His recent reseach interests include model reduction of kinetic equation, numerical method for fluid dynamics.