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In this article the Dark operators are defined to be kinetic energy operators on Hilbert spaces \(L^2(\mathbb {R}^d, \mu )\) , where \(\mu \) is a \(\sigma \) -finite positive measure singular with respect to the Lebesgue measure. The kinetic energy operators are defined through a quadratic form associated to \(\mu \) . In this paper we study the case when \(\mu \) are atomic measures. In some cases we identify the Dark Operators and obtain their spectra. We find surprising connections, in the case of counting measure to the adjacency operators on the lattice in some cases.

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Spectral Theory of Some Dark Operators

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  • 发表日期：

    2025年

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  • W. Kirsch,
  • M. Krishna

摘要

In this article the Dark operators are defined to be kinetic energy operators on Hilbert spaces \(L^2(\mathbb {R}^d, \mu )\) , where \(\mu \) is a \(\sigma \) -finite positive measure singular with respect to the Lebesgue measure. The kinetic energy operators are defined through a quadratic form associated to \(\mu \) . In this paper we study the case when \(\mu \) are atomic measures. In some cases we identify the Dark Operators and obtain their spectra. We find surprising connections, in the case of counting measure to the adjacency operators on the lattice in some cases.