Spectral force approach to solve the time-dependent Wigner-Liouville equation

The Wigner-Liouville (WL) equation is well suited to describe electronic transport in semiconductor devices. In the effective mass approximation the one dimensional WL equation reads ∂/∂t f(x, p, t) + p/m ∂/∂x f(x, p, t)-1/h
²
∫ dp' W(x, p-p')f(x, p', t) = 0; (1) with the Wigner kernel given by W(x, p) = -i/2π ∫ dx' exp (-i px'/h) [V (x + x'/2)-V (x-x'/2)].(2) The Wigner kernel introduces a non-local interaction with the potential V(x), in accordance with quantum theory. Unfortunately, even for this simple interaction the mathematical form includes a highly oscillatory component (exp [-i p·x/h]) which impedes stable numerical implementation based on finite differences or finite elements.