Wigner approach to the two-band Kane model for a tunneling diode

In this article we study a simple model for describing the electron behavior in a heterogeneous material characterized by two energy bands. The model, the so-called Kane system, is composed of two Schrodinger-like equations coupled by a k - P term which describes the interband tunneling. We give a physical justification of the Kane system and reformulate it in terms of Wigner functions. The well-posedness of the corresponding Wigner system is investigated in a suitable Hilbert space.