Parity-decomposition and Moment Analysis for Stationary Wigner Equation with Inflow Boundary Conditions

TL;DR

This paper analyzes the stationary Wigner equation with inflow boundary conditions using parity decomposition, proving well-posedness and uniqueness of solutions for even and odd parts without cutoff approximations, and demonstrates the approach with an example.

Contribution

It introduces a parity decomposition method for the stationary Wigner equation, establishing well-posedness and uniqueness results without cutoff approximations.

Findings

The even part's initial value problem is well-posed.

The odd part's solution is unique in the odd L^2 space.

An example illustrates the solution derivation using the analysis.

Abstract

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd $L^2$-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.