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

I. A. Sheipak

Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 302-308

Résumé

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 $d?$.

Détail

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