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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 6, Pages 258–273 (Mi izv8388)  

On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness

J. V. Tikhonov, I. A. Sheipak

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study spectral properties of the boundary-value problem
\begin{gather*} -y"-\lambda\rho y=0,
y(0)=y(1)=0, \end{gather*}
in the case when the weight $\rho$ belongs to the space $\mathcal M$ of multipliers from the space $\overset{\circ}{W} _2^1[0,1]$ to the dual space $(\overset{\circ}{W} _2^1[0,1])'$. We obtain a criterion for the generalized derivative (in the sense of distributions) of a piecewise-constant affinely self-similar function to lie in $\mathcal M$. For general weights in this class we show that the spectrum of the problem is discrete and the eigenvalues grow exponentially. The nature of this growth is determined by the parameters of self-similarity. When the parameters of self-similarity reach the boundary of the set where $\rho\in\mathcal M$, the problem exhibits continuous spectrum.

Keywords: self-similar functions, multipliers in Sobolev spaces, string equation, spectral asymptotics.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00705
13-01-12476
Russian Science Foundation 14-11-00754
The results of § 2 were obtained with the support of the Russian Foundation for Basic Research (grants no. 13-01-00705, 13-01-12476). The results of § 3 and § 4 were obtained with the support of the Russian Science Foundation (project no. 14-11-00754).


DOI: https://doi.org/10.4213/im8388

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English version:
Izvestiya: Mathematics, 2016, 80:6, 1242–1256

Bibliographic databases:

UDC: 517.984+517.518.26
MSC: 28A80, 34B24, 34L20, 46E35, 47E05
Received: 13.04.2015
Revised: 30.10.2015

Citation: J. V. Tikhonov, I. A. Sheipak, “On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness”, Izv. RAN. Ser. Mat., 80:6 (2016), 258–273; Izv. Math., 80:6 (2016), 1242–1256

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