论文标题
有效的质量定理具有Bloch模式交叉
Effective mass theorems with Bloch modes crossings
论文作者
论文摘要
我们研究了一个Schr {Ö} dinger方程，模拟了小波长的渐近状态中电子在晶体中的动力学，与晶体的特征尺度相当。使用Floquet Bloch分解，我们获得了时间平均能量密度极限的描述。假设初始数据是在高阶Sobolev空间上统一界限的，并且Bloch模式之间的交叉处在最坏的圆锥形时，我们会做出相当一般的假设。我们表明，尽管它们产生了奇异性，但锥形穿越并不能捕获能量，也不会阻止分散。我们还研究了当Bloch带之间存在一些退化的交叉点时可能发生的模式之间的相互作用。
We study a Schr{ö}dinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wavelength 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 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.
