Boyko O. Direct and inverse problems for Stieltjes strings

The thesis is devoted to investigation of direct and inverse spectral problems generated by the equation for amplitude vector of a Stieltjes string, i.e. of an elastic weightless thread bearing finite number of point masses. The case of absence of viscous friction is considered to which the Dirichlet and Neumann conditions correspond as well as the cases of point friction at one of the ends of the string or at an interior point. The case of a star graph composed by Stieltjes strings is also considered. Conditions are found for a set of complex numbers necessary and sufficient to be the spectrum of a boundary value problem generated by a Stieltjes string subject to point viscous friction at one of the ends and at an interior point. Methods are proposed of finding values of the point masses and of lengths of the intervals between them using the spectrum of vibrations of the string and its total length.Uniqueness of solution is investigated for such inverse problems. Also the inverse problem is solved for a star graph composed by Stieltjes strings.