INSTABILITY PHENOMENA FOR THE FOURIER COEFFICIENTS

Let P be an elliptic differential operator on a non-compact connected manifold X; suppose that both X and the coefficients of P are real ana lytic. Given a pair of open sets D and sigma in X with sigma subset of subset of D subset of subset of X, we fix a sequence {e(nu)} of solut ions of Pu = 0 in D which are pairwise orthogonal under integration ov er both D and sigma. By orthogonality is meant the orthogonality in th e corresponding Sobolev spaces; we also assume a completeness of the s ystem on sigma. For a fixed y is an element of X\<(sigma)over bar> den ote by k(nu)(y) the Fourier coefficients of a fundamental solution Phi (., y) of P with respect to the restriction of {e(nu)} to sigma. Suppo se K is a compact set in D\<(sigma)over bar>, and let f be a distribut ion with support on K. In this paper we show, under appropriate condit ions on K, that if the moments {f, k(nu)} decrease sufficiently rapidl y in a certain precise sense, then these moments vanish identically. I n the most favorable cases, it is then possible to conclude that f = 0 . This phenomenon was previously noticed by the first author and L. ZA LCMAN for analytic and harmonic moments of f.