By Inc. BarCharts

ISBN-10: 1572225297

ISBN-13: 9781572225299

Dynamics and the examine of movement.

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Phys. 142, no. 2, 217–244 (1991). : Semi-classical methods in solid state physics, Comm. Math. Phys. 116, 401–415 (1988). : Homogenization of Diﬀerential Operators and Integral Functionals. Springer Verlag, (1994). : Perturbation theory for linear operators, Springer-Verlag, Berlin (1966). : Floquet theory for partial diﬀerential equations, Operator Theory: Advances and Applications, 60, Birkh¨ auser Verlag, Basel (1993). : Some methods in the mathematical analysis of systems and their control, Science Press, Beijing, Gordon and Breach, New York (1981).

Homogenization of the Schr¨ odinger equation and eﬀective mass theorems, Comm. Math Phys. 1–22 (2005). 1–16 (2006). 36, Kluwer Academic Publishers, Dordrecht (1990). : Semi-classical limits in a crystal with a Coulombian selfconsistent potential: eﬀective mass theorems, Asymptot. Anal. 19, no. 95–116 (1999). : Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation, J. Comput. Phys. 213, no. 288–310 (2006). : Asymptotic analysis for periodic structures, North-Holland, Amsterdam (1978).

6. By changing the main assumption on the Bloch spectrum it is possible to obtain a fourth order homogenized equation instead of the usual Schr¨ odinger equation. Speciﬁcally, if we consider i 2 ∂u − div A x ∇u + −2 x x (79) (i) λn (θn ) is a simple eigenvalue, (ii) ∇θ λn (θn ) = 0, ∇θ ∇θ λn (θn ) = 0, ∇θ ∇θ ∇θ λn (θn ) = 0, (80) c + 2 u = 0, ∂t d x, and if we make the following assumption, instead of (62), then, for the same type of initial data (66), we can prove that the solution of (79) can be written as u (t, x) = ei λn (θ n )t 4 e2iπ θ n ·x x ψn ( , θn )v(t, x) + r (t, x), (81) + 2 N + 2 N where r converges strongly to 0 in L∞ loc R ; L (R ) and v ∈ C R ; L (R ) is the solution of the fourth-order homogenized problem 30 G.

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