Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues

Abstract

The oscillation equation for a singular string with discrete weight generated by a self-similar n n -link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an ( n − 1 ) (n-1) -periodic Jacobi matrix. In the case of n = 3 n=3 , a complete description of the spectrum of the problem is given, and a criterion for emergence of an eigenvalue in a gap of the continuous spectrum is obtained.