Boundary Inverse Problem for Star-Shaped Graph with Different Densities Strings-Edges

Abstract

The paper is devoted to the mathematical modelling of star-shaped geometric graphs with n rib-strings of dierent density and the solution of the boundary inverse spectral problem for Sturm-Liouville differential operators on these graphs. Earlier it was shown that if strings have the same length and densities, then the stiness coeffcients of springs at the ends of graph strings are not uniquely recovered from natural frequencies. They are found up to permutations of their places. We show, that if the strings have dierent densities, then the stiness coeffcients of springs on the ends of graph strings are uniquely recovered from all natural frequencies. Counterexamples are shown that for the unique recovery of the stiness coeffcients of springs on n dead ends of the graph, it is not enough to use n natural frequencies. Examples are also given showing that it is suffcient to use n + 1 natural frequencies for the uniqueness of the recovery of the stiness coeffcients of springs at the n ends of the strings. Those, the uniqueness or non-uniqueness of the restoration of the stiness coeffcients of springs at the ends of the strings depends on whether the string densities are identical or different.