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Mat. Zametki, 2015, Volume 97, Issue 2, Pages 302–308 (Mi mz10576)  

Asymptotics of the Spectrum of a Differential Operator with the Weight Generated by the Minkowski Function

I. A. Sheipak

M. V. Lomonosov Moscow State University

Abstract: This paper is devoted to the study of the asymptotics of the spectrum of the boundary-value problem
$$ -y"-\lambda\rho y=0, \qquad y(0)=y(1)=0, $$
where $\rho$ is the generalized derivative of the Minkowski function, i.e., $\rho=?'(x)$ (here $?(x)$ is the “question-mark function” first defined by Minkowski, who introduced this notation). For the eigenvalues of the problem, asymptotic two-sided estimates of power type are obtained. The order of the power is determined by the Hausdorff dimension of the support of the Minkowski measure $d?$.

Keywords: spectrum of a differential operator, Minkowski function, boundary-value problem, Hausdorff dimension, Minkowski measure.

Funding Agency Grant Number
Russian Science Foundation 14-11-00754
This work was supported by the Russian Science Foundation (grant no. 14-11-00754).


DOI: https://doi.org/10.4213/mzm10576

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English version:
Mathematical Notes, 2015, 97:2, 289–294

Bibliographic databases:

Document Type: Article
UDC: 517.984
Received: 22.09.2014

Citation: I. A. Sheipak, “Asymptotics of the Spectrum of a Differential Operator with the Weight Generated by the Minkowski Function”, Mat. Zametki, 97:2 (2015), 302–308; Math. Notes, 97:2 (2015), 289–294

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