Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer; Tony Samuel; Hendrik Weyer

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 3, page 313-335
  • ISSN: 0010-2628

Abstract

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Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure η supported on a compact subset of ℝ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator ∇ η and a second order differential operator Δ η , with respect to η . We generalize this approach to measures of the form η : = ν + δ , where ν is non-atomic and δ is finitely supported. We determine analytic properties of ∇ η and Δ η and show that Δ η 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 Δ η . For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

How to cite

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Kesseböhmer, Marc, Samuel, Tony, and Weyer, Hendrik. "Measure-geometric Laplacians for partially atomic measures." Commentationes Mathematicae Universitatis Carolinae 61.3 (2020): 313-335. <http://eudml.org/doc/297013>.

@article{Kesseböhmer2020,
abstract = {Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $\eta $ supported on a compact subset of $\mathbb \{R\}$ U. Freiberg and M. Zähle 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.},
author = {Kesseböhmer, Marc, Samuel, Tony, Weyer, Hendrik},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Kreĭn--Feller operator; spectral asymptotics; harmonic analysis},
language = {eng},
number = {3},
pages = {313-335},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Measure-geometric Laplacians for partially atomic measures},
url = {http://eudml.org/doc/297013},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Kesseböhmer, Marc
AU - Samuel, Tony
AU - Weyer, Hendrik
TI - Measure-geometric Laplacians for partially atomic measures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 3
SP - 313
EP - 335
AB - Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $\eta $ supported on a compact subset of $\mathbb {R}$ U. Freiberg and M. Zähle 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.
LA - eng
KW - Kreĭn--Feller operator; spectral asymptotics; harmonic analysis
UR - http://eudml.org/doc/297013
ER -

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