| Title:
|
Measure-geometric Laplacians for partially atomic measures (English) |
| Author:
|
Kesseböhmer, Marc |
| Author:
|
Samuel, Tony |
| Author:
|
Weyer, Hendrik |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
313-335 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Kreĭn--Feller operator |
| Keyword:
|
spectral asymptotics |
| Keyword:
|
harmonic analysis |
| MSC:
|
35P20 |
| MSC:
|
42B35 |
| MSC:
|
47G30 |
| idZBL:
|
Zbl 07286007 |
| idMR:
|
MR4186110 |
| DOI:
|
10.14712/1213-7243.2020.026 |
| . |
| Date available:
|
2020-11-27T07:39:48Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148469 |
| . |
| Reference:
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