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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.

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
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