Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions
Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions
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贝特-索末菲阈值的上限和二维泊松随机哈密顿量的谱
DOI：
10.1007/s00023-012-0180-1
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
T. Mine
中科院分区：
文献类型：
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作者：
M. Kaminaga;T. Mine
我们考虑具有局部平方可积周期势V的Schrödinger算子，给出了关于基本区域上V的平方可积范数的Bethe-Sommerfeld阈值(其上没有谱间隙的最小能量)的一个上界。作为应用，我们证明了具有泊松随机势的二维薛定谔算符的谱几乎必然等于正实轴或整个实轴，无论单位势的负部分是否等于零。后一个结果补充了Ando等人的结果缺失的部分。(Ann Henri Poincaré7：145-160,2006)。
We consider the Schrödinger operator onwith a locally square-integrable periodic potentialVand give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm ofVon a fundamental domain, provided thatVis small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006).