ИСТИНА

We consider Sturm--Liouville problem with a weight from the Sobolev space with a negative index of smoothness. If weight is a $n$-term self-similar non-compact multiplier the spectral problem for the string is equivalent to the spectral problem for the $(n-1)$-periodic Jacobian matrix. In the case $n=3$ a complete description of the spectrum of the problem is given, and a criterion for the appearance of an eigenvalue in the gap of the continuous spectrum is obtained. In the general situation $n\geqslant 3$, it is shown that the spectrum consists of $n-1$ segments of the continuous spectrum (some of which may touch) in the gaps between which there can be no more than $n-2$ eigenvalues (no more than one in each interval). We consider the equation of oscillation of a singular string with a weight from the Sobolev space with a negative index of smoothness. If weight is a $n$-term self-similar non-compact multiplier the spectral problem for the string is equivalent to the spectral problem for the $(n-1)$-periodic Jacobian matrix. In the case $n=3$ a complete description of the spectrum of the problem is given, and a criterion for the appearance of an eigenvalue in the gap of the continuous spectrum is obtained. In the general situation $n\geqslant 3$, it is shown that the spectrum consists of $n-1$ segments of the continuous spectrum (some of which may touch) in the gaps between which there can be no more than $n-2$ eigenvalues (no more than one in each interval).