Homogenization limits and Wigner transforms. (English) Zbl 0881.35099

Gérard, Patrick; Markowich, Peter A.; Mauser, Norbert J.; Poupaud, Frédéric

Commun. Pure Appl. Math. 50, No. 4, 323-379 (1997).

This paper deals with a theory for carrying out homogenization limits for quadratic functions \(n^\varepsilon=|u^\varepsilon(t, x)|^2\) of solutions \(u^\varepsilon\) of the following type Cauchy problems: \[ \varepsilon u^\varepsilon_t+ P^\varepsilon u^\varepsilon=0,\quad u^\varepsilon|_{t=0}= u^\varepsilon_I(x), \] where \(\varepsilon>0\) is a small parameter, \(\varepsilon\to 0\), \(u^\varepsilon\in L^2(\mathbb{R}^m_x)\), and \(P^\varepsilon\) is an anti-selfadjoint spatial pseudodifferential operator. To do this, the authors introduce a special phase space – the space of Wigner measures and calculate them by solving some kinetic equations. The weak limits of \(n^\varepsilon\) are obtained by taking moments of the Wigner measure. Applications are given to the Schrödinger equation, to the acoustic equation in a periodic medium, and to the Dirac equation.

Reviewer: P.Popivanov (Sofia)

MSC:

35Q40	PDEs in connection with quantum mechanics
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

homogenization limit; Wigner measure; Schrödinger equation; acoustic equation; Dirac equation