Место издания:Издательство Ростовского отделения Российской инженерной академии г. Ростов
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Аннотация:We study spectral properties of boundary value problem
$$
-y''=\lambda\rho y,
$$
$$
y(0)=y(1)=0,
$$
in the case where the weight $\rho$ is a multiplier from the space
${{\raisebox{0.2ex}{\(\stackrel{\circ}{W}\)}}{_2^{1}[0,1]}}$ into the dual space ${{\raisebox{0.2ex}{\(\stackrel{\circ}{W}\)}}{_2^{-1}[0,1]}}$.
We have received the necessary and sufficient conditions under which a self-similar function generates multiplier in these spaces.
The class of compact self-similar multipliers was described. For such weights-multipliers the spectrum of the problem is discrete and eigenvalues have exponentially growth. Characteristics of growth are determined by the parameters of self-similarity. We have constructed the class of non-compact multipliers, for which the spectrum of the problem is continuous. The full description of continuous spectrum is obtained for self-similar weights based on two subintervals.