Convergence of Semi-discrete Stationary Wigner Equation with Inflow Boundary Conditions

TL;DR

This paper introduces a new semi-discretization method for the stationary Wigner equation with inflow boundary conditions, ensuring well-posedness and convergence, supported by analysis of solution regularity.

Contribution

It proposes a novel semi-discretization approach using the Whittaker-Shannon interpolation formula, establishing convergence and regularity results for the stationary Wigner equation.

Findings

The semi-discrete scheme is well-posed.

Solutions of the discrete problem converge to the continuous solution.

Regularity of the continuous solution is characterized.

Abstract

Making use of the Whittaker-Shannon interpolation formula with shifted sampling points, we propose in this paper a well-posed semi-discretization of the stationary Wigner equation with inflow BCs. The convergence of the solutions of the discrete problem to the continuous problem is then analysed, providing certain regularity of the solution of the continuous problem.