ИСТИНА

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.