分形Turán-Nazarov不等式与薛定谔方程的可观测性
Fractal Turán-Nazarov Inequality and Observability for Schrödinger Equations
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中文总结 AI 辅助
研究分形集上薛定谔方程可观测性不等式与唯一延拓性的局限。先将经典Turán-Nazarov不等式推广到分形环境，通过得到的精确界构造反例，证明观测集为分形时薛定谔方程的可观测性和唯一延拓性可能失效。
中文摘要 AI 辅助
本文建立了分形集上薛定谔方程可观测性不等式和唯一延拓性的局限性。我们证明，与热方程不同，这些性质在分形环境中可能不成立。为此，我们首先将经典的Turán-Nazarov不等式（它给出了正测度集上形如\(\sum_{k = 1}^n c_k e^{2\pi i m_k t}\)的三角多项式的下界）推广到分形环境。与经典情况不同，不等式中的常数在次数\(n\)上失去了一致性，我们得到了依赖于\(n\)和频率差\(m_n - m_1\)的精确界。这些改进使我们能够构造明确的反例，表明当观测集是分形时，薛定谔方程的可观测性和唯一延拓性可能不成立。
英文摘要
This paper establishes limitations on observability inequality and unique continuation for Schrödinger equations on fractal sets. We prove that, in contrast to the heat equation, such properties can fail in fractal settings. To achieve this, we first extend the classical Turán--Nazarov inequality, which provides lower bounds of trigonometric polynomials of the form $\sum_{k=1}^nc_ke^{2πim_kt}$ on sets of positive measure, to the fractal setting. Unlike in the classical case, the constant in the inequality loses uniformity in the degree $n$, and we obtain sharp bounds depending on both $n$ and the frequency difference $m_n-m_1$. These refinements then enable us to construct explicit counterexamples, showing that observability and unique continuation may fail for Schrödinger equations when the observation set is fractal.
