Hybrid coupling of a one-dimensional energy-transport Schrödinger system (Q1682384)

This paper refers to the problem of energy transport in quantum physics. It considers the domain \([0,L]\) of the one-dimensional case. The quantum zone \(Q=[x_{1},x_{2}]\) is described by the Schrödinger equation \[ -\frac{1}{2m}\partial_{xx}\psi_{k}+V(x)= e(x_{1},k)\psi_{k},\qquad k>0, \] \[ -\frac{1}{2m}\partial_{xx}\psi_{k}+V(x)= e(x_{2},k)\psi_{k},\qquad k<0, \] where \(m\) is the mass and \(e\) is the total particle energy of an electron. This system is complemented with the boundary conditions at the points \(x_{1}\) and \(x_{2}\). The classical zone \(C=[0,x_{1}]\bigcup[x_{2},L]\) is modeled by the Boltzmann equation, complemented with the boundary conditions at points \( x=0\) and \(x=L \). The numerical simulation of these problems is given using the Crank-Nicolson scheme in the quantum zone and the finite element method in the classical zone.