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arXiv 1106.0632math.AP
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位势为W^{ε，p}的二维薛定谔方程反问题的稳定性和唯一性

Stability and uniqueness for the inverse problem of the Schrödinger equation in 2D with potentials in W^{ε,p}

Eemeli Blåsten

备注：
20 pages, to appear in my PhD thesis
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摘要：

这个结果将作为我的博士论文的一部分，经过一些精简后发表。这份手稿包含了这一说法的证据，但没有经过同行评审。证明了二维薛定谔方程的反问题在势函数满足W^{ε，p}（Ω），ε>0，p>2时的唯一性和稳定性.证明的想法是使用Bukhgeim的振荡指数解。利用Alessandrini恒等式和定态相位，我们从Dirichlet-Neumann映射的差分中得到了关于势差的信息。使用插值，我们看到最差的余项以1 - ε - β的指数衰减。这里β是我们在共轭柯西算子的范数估计中得到的指数。我们使它任意接近1，因此当ε > 0时存在唯一性和稳定性。

英文摘要：

This result will be published as part of my PhD thesis after some streamlining. This manuscript contains the proof of the claim, but is not peer-reviewed. We prove uniqueness and stability for the inverse problem of the 2D Schrödinger equation in the case that the potentials give well posed direct problems and are in W^{ε,p}(Ω), ε>0, p>2. The idea of the proof is to use Bukhgeim's oscillating exponential solutions. By Alessandrini's identity and stationary phase we get information about the difference of the potentials from the difference of the Dirichlet-Neumann maps. Using interpolation, we see that the the worst of the remainder terms decays with an exponent of 1 - ε - β. Here β is the exponent which we get in a norm estimate for the conjugated Cauchy operator. We get it arbitrarily close to 1, so there is uniqueness and stability when ε > 0.

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