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arXiv 2201.05109math.APmath-phmath.MP
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非线性薛定谔方程解从零点的唯一性

Uniqueness of solutions to nonlinear Schrödinger equations from their zeros

Christoph Kehle, João P. G. Ramos

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30 pages, 0 figures
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摘要：

我们展示了新类型的唯一性和刚性结果的薛定谔方程的非线性情况下，或在存在一个复值的潜力。作为我们的主要结果，我们得到的平凡解$u=0$是唯一的解决方案，其中假设$u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$举行，$D\subset \mathbb{R}^d$是某些子集的余维1。特别地，$D$对于维度$d=1$是离散的。我们的主要定理可以看作是离散傅立叶唯一性对的非线性模拟，如著名的Radchenko-Viazovska公式，以及第二作者和M.索萨的整数幂。作为一个补充应用，我们推导了一些半线性椭圆型方程解从零点开始的刚性结果。

英文摘要：

We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only solution for which the assumptions $u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$ hold, where $D\subset \mathbb{R}^d$ are certain subsets of codimension one. In particular, $D$ is discrete for dimension $d=1$. Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko--Viazovska formula, and the uniqueness result of the second author and M. Sousa for powers of integers. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros.

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