MULTIDIMENSIONAL INVERSE PROBLEM WITH INCOMPLETE BOUNDARY SPECTRAL DATA

We consider an inverse boundary problem for a general second order sel f-adjoint elliptic differential operator on a compact differential man ifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the opera tor to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical o bject in the orbit of the group. This object, the canonical Schrodinge r operator, is uniquely determined via its incomplete boundary spectra l data.