Discontinuous Density Function Identification

Abstract

The work is devoted to the identification step density function of a string. The inverse problem consistsof recovering constant densities ρi of eigenvalue problem. It is shown that if we use only the natural frequenciesof the boundary value problem itself to restore the step density, then this inverse problem has an infinite number ofsolutions ρ = (ρ1, ρ2, . . . , ρn) in $mathrm R^{mathrm n}$and unique solution in a sufficiently small area Ω ⊂ $mathrm R^{mathrm n}$. For the uniqueness of therecovery of the step density of a string, the natural frequencies of one boundary value problem are not enough. Weneed to use the natural frequencies of the two boundary problems. To uniquely reconstruct a step density function,we need to use natural frequencies of the boundary value problem itself and natural frequencies of another boundaryproblem, which differs from the first one only by one boundary condition. In M. Krein uniqueness theorems, torestore the continuous density function, we used all the eigenvalues of the two problems. In contrast to the M. Kreinuniqueness theorems, for the uniqueness of the recovery of the n-step density function, we need a finite number ofeigenvalues.