多维周期Schrodinger算子的微扰理论与Bethe-Sommerfeld猜想
Perturbation Theory for the Periodic Multidimensional Schrodinger Operator and the Bethe-Sommerfeld Conjecture
摘要：
本文给出了任意维周期薛定谔算子的Bloch本征值和Bloch函数的任意阶渐近公式，当相应的准动量位于衍射超平面附近时.此外，将我以前的文章中得到的Bloch本征值和Bloch函数在相应的拟动量远离衍射超平面时的渐近公式作了改进和扩大，得到了具有周期势的多维Schrodinger算子的完全微扰理论.此外，我们估计了高能区的等能面的测度，这意味着Bethe-Sommerfeld猜想对任意维数和任意晶格的有效性。
英文摘要：
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator 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 quasimomentum 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 energy region which implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimension and arbitrary lattice.
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