Effective mass theorems with Bloch modes crossings. (English) Zbl 1502.81032
Summary: We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 82D25 | Statistical mechanics of crystals |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 30H30 | Bloch spaces |
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