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
$\bigl(\overset{\circ}{W}{}_2^1[0,1]\bigr)'$. 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.