This article is cited in 2 papers

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.

UDC: 517.984

Received: 22.09.2014

DOI: 10.4213/mzm10576

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

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