On recovering of the density in the plane domain from incomplete spectral data

An inverse spectral problem of the density recovering is considered. The first $N$ eigenvalues and traces on the boundary of the normal derivatives of eigenfunctions for Dirichlet problem are chosen as input data. The theorem that an error in the density approximation is less than $cN^{-\beta}$, with some positive constants $c$ and $\beta$ is proved.