Perturbation Theory for the Periodic Multidimensional Schrodinger Operator and the Bethe-Sommerfeld Conjecture
Perturbation Theory for the Periodic Multidimensional Schrodinger Operator and the Bethe-Sommerfeld Conjecture
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周期性多维薛定谔算子的微扰理论和贝特-索末菲猜想
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发表时间：
2006
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通讯作者：
O. Veliev
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文献类型：
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作者：
O. Veliev
本文得到了任意维的周期薛定谔算子- - Δ+ q(x)的Bloch特征值和Bloch函数在对应的准动量位于衍射超平面附近时的任意阶的渐近公式。此外，通过对前人所得的Bloch本征值和Bloch函数的渐近公式进行改进和扩大，得到了具有周期势的多维薛定谔算子的完全摄动理论。此外，我们估计了高能区等能表面的测量值，这表明贝特-索默菲尔德猜想对于任意维数和任意晶格是有效的。
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger opera- tor − Δ+ q(x), of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane. Besides, writing the asymptotic formulas for the Bloch eigenvalue and the Bloch function, when corresponding quasi- momentum lies far from the diffraction hyperplanes, obtained in my previous papers in improved and enlarged form, we obtain the complete perturbation theory for the multidimensional Schrodinger operator with a periodic potential. Moreover, we estimate the measure of the isoenergetic surfaces in the high en- ergy region which implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimension and arbitrary lattice.