论文标题
在蒙哥马利家族四分之一振荡器家族的特征值的关键点上
On critical points of eigenvalues of the Montgomery family of quartic oscillators
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
We discuss spectral properties of the family of quartic oscillators $\mathfrak h_{\mathcal M}(α) =-\frac{d^2}{dt^2} +\Big(\frac{1}{2} t^{2} -α\Big)^2$ on the real line, where $α\in \mathbb{R}$ is a parameter. This operator appears in a variety of applications coming from quantum mechanics to harmonic analysis on Lie groups, Riemannian geometry and superconductivity. We study the variations of the eigenvalues $λ_j(α)$ of $\mathfrak h_{\mathcal M}(α)$ as functions of the parameter $α$.We prove that for $j$ sufficiently large, $α\mapsto λ_j(α)$ has a unique critical point, which is a nondegenerate minimum.We also prove that the first eigenvalue $λ_1(α)$ enjoys the same property and give a numerically assisted proof that the same holds for the second eigenvalue $λ_2(α)$. The proof for excited states relies on a semiclassical reformulation of the problem. In particular, we develop a method permitting to differentiate with respect to the semiclassical parameter, which may be of independent interest.
