  We explore the possibilities of reaching the characterization of
eigenfunction of Laplacian as a degenerate case of the inverse Paley-Wiener
theorem (characterizing functions whose Fourier transform is supported on a
compact annulus) for the Riemannian symmetric spaces of noncompact type. Most
distinguished prototypes of these spaces are the hyperbolic spaces. The
statement and the proof of the main result work mutatis-mutandis for a number
of spaces including Euclidean spaces and Damek-Ricci spaces.
