Numerical Solution of the Constrained Wigner Equation

Quantum electron transport in modern semiconductor devices can be described by a Wigner equation which is formally similar to the classical Liouville equation. The stationary Wigner equation has a singularity at zero momentum (k=0). In order to get a non-singular solution it is necessary to impose a constraint for the solution at k=0 which gives the constrained Wigner equation. We introduce a Petrov-Galerkin method for the solution of the corresponding constrained sigma equation. The constraint in the Wigner equation is interpreted as an extra test function and is naturally incorporated in the method.