We establish a sign correlation universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a trigonometric asymptotic law. Our results are inspired by the classical WKB asymptotic approximation for Sturm–Liouville operators, and in particular we obtain non-trivial sign correlations for eigenfunctions of generic Schrödinger operators (including the harmonic oscillator), as well as Laguerre and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x \neq y$, $$\frac{1}{3} \leq \lim_{N \to \infty}{ \frac{1}{N} \# \{0 \leq n < N\colon \mathrm {sgn}(w_n(x)) = \mathrm {sgn}(w_n(y)) \}} \leq \frac{2}{3},$$ and that these bounds are optimal, and can be attained. Our methods extend to other problems of similar flavor and we also discuss a number of open problems.

中文翻译：

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微分算子本征函数符号相关的普遍性规律

我们为满足三角渐近律的函数序列 $\{w_n\}_{n \in \mathbb{N}}$ 建立了符号相关普遍性定律。我们的结果受到 Sturm-Liouville 算子的经典 WKB 渐近近似的启发，特别是我们获得了泛型薛定谔算子（包括谐振子）以及 Laguerre 和 Chebyshev 多项式的特征函数的非平凡符号相关性。给定两个不同的点 $x, y \in \mathbb{R}$，我们询问 $w_n(x)$ 和 $w_n(y)$ 多久具有相同的符号。渐近地，人们会期望这有一半的时间是正确的，但事实证明并非总是如此。在某些自然假设下，我们证明，对于所有 $x \neq y$，$$\frac{1}{3} \leq \lim_{N \to \infty}{ \frac{1}{N} \# \{0 \leq n < N\colon \mathrm {sgn}(w_n(x)) = \mathrm {sgn}(w_n(y)) \}} \leq \frac{2}{3},$$ 并且这些界限是最优的，并且可以达到。我们的方法扩展到其他类似的问题，我们还讨论了一些开放的问题。