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AI InsightKrein uniqueness theorems, for the uniqueness of the recovery of the n-step density function, we need a finite number of eigenvalues.
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Abstract
The work is devoted to the identification step density function of a string. The inverse problem consists of recovering constant densities $ \rho_{i}$ of eigenvalue problem. It is shown that if we use only the natural frequencies of the boundary value problem itself to restore the step density, then this inverse problem has an infinite number of solutions $ \rho = \left( \rho_{1}, \rho_{2}, \dots , \rho_{n} \right) $ in $ {\mathbb{R}}^{n} $ and unique solution in a sufficiently small area $ \Omega \subset \mathbb{R}^{n}$. For the uniqueness of the recovery of the step density of a string, the natural frequencies of one boundary value problem are not enough. We need 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 boundary problem, which differs from the first one only by one boundary condition. In M. Krein uniqueness theorems, to restore the continuous density function, we used all the eigenvalues of the two problems. In contrast to the M. Krein uniqueness theorems, for the uniqueness of the recovery of the n-step density function, we need a finite number of eigenvalues.