In this paper we develop an in‐depth theoretical investigation of the discrete Hamiltonian eigenbasis, which remains quite unexplored in the geometry processing community. This choice is supported by the fact that Dirichlet eigenfunctions can be equivalently computed by defining a Hamiltonian operator, whose potential energy and localization region can be controlled with ease. We vary with continuity the potential energy and study the relationship between the Dirichlet Laplacian and the Hamiltonian eigenbases with the functional map formalism. We develop a global analysis to capture the asymptotic behavior of the eigenpairs. We then focus on their local interactions, namely the veering patterns that arise between proximal eigenvalues. Armed with this knowledge, we are able to track the eigenfunctions in all possible configurations, shedding light on the nature of the functional maps. We exploit the Hamiltonian‐Dirichlet connection in a partial shape matching problem, obtaining state of the art results, and provide directions where our theoretical findings could be applied in future research.

中文翻译：

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离散哈密顿泛函映射的参数分析

在本文中，我们对离散哈密顿本征基进行了深入的理论研究，这在几何处理社区中仍未得到探索。这一选择得到以下事实的支持：狄利克雷特征函数可以通过定义哈密顿算符来等效计算，其势能和定位区域可以轻松控制。我们随着势能的连续性而变化，并用泛函映射形式研究Dirichlet Laplacian 和Hamiltonian 特征库之间的关系。我们开发了一个全局分析来捕捉特征对的渐近行为。然后我们关注它们的局部相互作用，即在近端特征值之间出现的转向模式。有了这些知识，我们就能够在所有可能的配置中跟踪特征函数，阐明功能图的性质。我们在部分形状匹配问题中利用哈密顿-狄利克雷连接，获得了最先进的结果，并提供了我们的理论发现可以应用于未来研究的方向。