We test M. Berry’s ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions (“boundary-adapted arithmetic random waves”). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac–Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of “bad” subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the “spectral semi-correlations”, a concept joining the likes of “spectral correlations” and “spectral quasi-correlations” in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.

中文翻译：

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算术随机波的节点长度的边界效应和光谱半相关

我们在存在边界的情况下在节点缺陷上测试 M. Berry 的 ansatz。研究方台球，其中高光谱简并性允许引入随机拉普拉斯本征函数（“边界适应算术随机波”）的高斯系综。作为精确渐近分析的结果，在沿着能级的一般序列的高能量极限中推导出预期节点长度的渐近扩展中的两个项。发现节点长度的精确节点不足或过剩以明确的方式取决于能级的算术性质。为了获得上述结果，我们应用 Kac-Rice 方法来计算高斯随机场的预期节点长度。这样的应用程序揭示了主要障碍，例如“坏”子域的出现，人们希望，对节点长度的贡献微不足道。幸运的是，我们能够将这种贡献减少到计算“光谱半相关”的数论问题上，该概念将“光谱相关”和“光谱准相关”等概念结合在一起，对算术的节点长度产生影响动力系统。这项工作依赖于 J. Bourgain 的几项突破性技术，他对该主题的兴趣有助于在很大程度上塑造它，并且他与 E. Bombieri 联合在光谱相关方面的基础工作对该领域产生了至关重要的影响。将“谱相关”和“谱准相关”之类的概念结合在一起，对算术动力系统的节点长度产生影响。这项工作依赖于 J. Bourgain 的几项突破性技术，他对该主题的兴趣有助于在很大程度上塑造它，并且他与 E. Bombieri 联合在光谱相关方面的基础工作对该领域产生了至关重要的影响。将“谱相关”和“谱准相关”之类的概念结合在一起，对算术动力系统的节点长度产生影响。这项工作依赖于 J. Bourgain 的几项突破性技术，他对该主题的兴趣有助于在很大程度上塑造它，并且他与 E. Bombieri 联合在光谱相关方面的基础工作对该领域产生了至关重要的影响。