长盒上的随机Schrodinger算子、噪声爆炸和GOE
Random Schrodinger operators on long boxes, noise explosion and the GOE
摘要：
证明了d ≥ 2维随机薛定谔算子在局域化跃迁处的本征值与高斯正交包络（GOE）的本征值相似.我们证明了存在n乘m个盒子的序列，其中1<<m<<n，使得低无序的本征值收敛于GOE的极限本征值过程Sine1。对于GOE的情况下，这是第一个例子，维格纳的著名的预测是严格证明：我们展示了一个复杂的系统，其特征值的行为像随机矩阵。
英文摘要：
It is conjectured that the eigenvalues of random Schrodinger operators at the localization transition in dimensions d>=2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n by m boxes with 1<<m<<n so that the eigenvalues in low disorder converge to Sine1, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.
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