Scattering and spectral problems for perturbations of multidimensional biharmonic operator

The subject of this work concerns to the classical inverse scattering and inverse spectral problems. The inverse scattering problem can be formulated as follows: do the knowledge of the far field pattern uniquely determines (and how) the unknown coefficients of given differential operator? Saito's formula and uniqueness result, as well as the reconstruction of singularities, are obtained for the scattering problems (see [1], [2]). The inverse spectral problem can be formulated as follows: do the Dirichlet eigenvalues and the derivatives (of which order?) of the normalized eigenfunctions at the boundary determine uniquely the coefficients of the corresponding differential operator? In the present work we show that the knowledge of the discrete Dirichlet spectrum and some special derivatives up to the third order of the normalized eigenfunctions at the boundary uniquely determine the coefficients of the operator of order 4 which is the second order perturbation of the biharmonic operator (see [3]). Usually in the literature is assumed the knowledge of the Dirichlet-to-Neumann map which uniquely determines the unknown coefficients. In the comparison with this we prove (in addition) that the Dirichlet-to-Neumann map can be uniquely determined by the spectral data.