Gelfʹand Inverse Problem for a Quadratic Operator Pencil

In this paper we consider the inverse boundary value problem for the operator pencil A(λ)=a(x, D)−iλb0(x)−λ2 where a(x, D) is an elliptic second-order operator on a differentiable manifold M with boundary. The manifold M can be interpreted as a Riemannian manifold (M, g) where g is the metric generated by a(x, D). We assume that the Gelʹfand data on the boundary is given; i.e., we know the boundary ∂M and the boundary values of the fundamental solution of A(λ), namely, Rλ(x, y), x, y∈∂M, λ∈C. We show that if (M, g) satisfies some geometric condition then the Gelʹfand data determine the manifold M, the metric g, the coefficient b0(x) uniquely and also the equivalence class of a(x, D) with respect to the group of generalized gauge transformations.