Abstract: Wigner transforms have been proven quite powerful for certain classes of homogenization problems with a distinguished scale of oscillations, in particular for (semi)classical limits of time dependent problems. Known to physicists for a long time as one of the most appealing "phase space formulations of quantum mechanics", the mathematically rigorous theory of Wigner measures, Wigner series and Wigner Bloch series has been developed in the last decade by P. Gerard, P.L. Lions, P.A. Markowich, N.J. Mauser, T. Paul, F. Poupaud and others.
In this talk we present an introduction of the method and applications for (semi)classical limits of linear and weakly nonlinear Schroedinger equations, eventually in the setting of a crystal, and we discuss the limitations of the method e.g. for strongly nonlinear problems. In this talk we present an introduction of the method and applications for (semi)classical limits of linear and weakly nonlinear Schroedinger equations, eventually in the setting of a crystal, and we discuss the limitations of the method e.g. for strongly nonlinear problems.
[BeMP2] P. Bechouche, N. J. Mauser and F. Poupaud: Semiclassical Limit for the Schrodinger-Poisson Equation in a Crystal, Comm. Pure and Appl. Math 54 (2001) 1–40
[GMMP] Gerard, P., Markowich, P.A., Mauser N. J., Poupaud, F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 321–377
[LiPa] Lions, P. L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993), 553–618
[ZhaZheMau] Zhang, P., Zheng, Y., Mauser, N. J.: The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. Pure Appl. Math. 55 (2002) 582–632