Kesseböhmer, Marc; Samuel, Tony; Weyer, Hendrik Measure-geometric Laplacians for partially atomic measures. (English) Zbl 1474.47092 Commentat. Math. Univ. Carol. 61, No. 3, 313-335 (2020). Summary: Motivated by the fundamental theorem of calculus, and based on the works of W. Feller [Ill. J. Math. 1, 459–504 (1957; Zbl 0077.29102)] as well as I. S. Kac and M. G. Kreĭn [“Criteria for the discreteness of the spectrum of a singular string”, Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 2(3), 136–153 (1958; Zbl 1469.34111)], given an atomless Borel probability measure \(\eta\) supported on a compact subset of \(\mathbb{R}\) U. Freiberg and M. Zähle [Potential Anal. 16, No. 3, 265–277 (2002; Zbl 1055.28002)] introduced a measure-geometric approach to define a first order differential operator \(\nabla_{\eta}\) and a second order differential operator \(\Delta_{\eta}\), with respect to \(\eta\). We generalize this approach to measures of the form \(\eta :=\nu+\delta\), where \(\nu\) is non-atomic and \(\delta\) is finitely supported. We determine analytic properties of \(\nabla_{\eta}\) and \(\Delta_{\eta}\) and show that \(\Delta_{\eta}\) is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of \(\Delta_{\eta}\). For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function. MSC: 47G30 Pseudodifferential operators 42B35 Function spaces arising in harmonic analysis 35P20 Asymptotic distributions of eigenvalues in context of PDEs Keywords:Kreĭn-Feller operator; spectral asymptotics; harmonic analysis Citations:Zbl 0077.29102; Zbl 1055.28002; Zbl 1469.34111 PDFBibTeX XMLCite \textit{M. Kesseböhmer} et al., Commentat. Math. Univ. 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