关于一般随机系数的若干反问题的注记
A note on several inverse problems with generally random coefficients
AI总结：
本文研究椭圆方程中随机系数的反问题，比较了从Dirichlet-to-Neumann映射的全律、期望、有限边界矩以及平均内部Green算子中可恢复的信息，得到了正反两方面结果。
摘要：
我们考虑椭圆方程的若干反问题，其系数是随机的，且不对随机性施加特殊的概率结构。主体部分处理Schrödinger方程。我们比较了从Dirichlet-to-Neumann映射的全律、其期望、其边界双线性形式的有限个联合矩以及平均内部Green算子中可恢复的信息。我们得到了正反两方面结果。Dirichlet-to-Neumann映射的全律确定随机势的律几乎是平凡的。然而，期望的Dirichlet-to-Neumann映射，更一般地，其边界矩的任何固定有限层次甚至不需要确定平均势。相反，平均Schrödinger Green算子确定势的点态均值和方差。在一个双原子模型中，它确定两点律的所有点态矩。附录包含电导率方程的相应结果。
英文摘要：
We consider several inverse problems for elliptic equations whose coefficients are random, without imposing a special probabilistic structure on the randomness. The main body treats the Schrödinger equation. We compare what can be recovered from the full law of the Dirichlet-to-Neumann map, from its expectation, from finitely many joint moments of its boundary bilinear form, and from the averaged interior Green's operator. We obtain both positive and negative results. That the full law of the Dirichlet-to-Neumann map determines the law of the random potential is almost trivial. However, the expected Dirichlet-to-Neumann map and, more generally, any fixed finite hierarchy of its boundary moments need not determine even the mean potential. In contrast, the averaged Schrödinger Green's operator determines the pointwise mean and variance of the potential. In a two-atom model it determines all pointwise moments of the two-point law. The appendices contain the corresponding results for the conductivity equation.
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