Gelf’and Inverse Problem for a Quadratic Operator Pencil

where [g]j,l=1 defines a C ∞-smooth Riemannian metric and b = (b1, ..., bm) and q are, correspondingly, C∞-smooth complex-valued 1-form and function on M . σ is a C∞-smooth complex-valued function on ∂M and ∂ν stands for the normal derivative. Let Rλ be the resolvent of (1), (2) which is meromorphic for λ ∈ C (see Sect. 3 and [1]) and let Rλ(x, y) be its Schwartz kernel. A natural analog of the Gel’fand inverse problem [2] is Problem I. Let ∂M and Rλ(x, y);λ ∈ C, x, y ∈ ∂M be given. Do these data (Gel’fand boundary spectral data, GBSD) determine (M,a(x,D), b0, σ) uniquely? Remark 1. Let Gλ be the Neumann-to-Dirichlet map Gλf := ufλ|∂M where A(λ)u (λ) = 0, Bu (λ) = f. (3)