Homogenization limits and Wigner transforms. (English) Zbl 0881.35099

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.


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
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