A REPRESENTATION THEOREM FOR SOLUTIONS OF SCHRODINGER TYPE EQUATIONS ON NONCOMPACT RIEMANNIAN-MANIFOLDS

Let X be a real analytic Riemannian manifold with a boundary partial derivative X. Denote its interior by X and its metric by g. Introduce on X a conformal metric h defined by h = p-2g where p(x) is a real analytic on X such that p(x) > 0 in X, p(x) = 0 and dp not-equal 0 on partial derivative X. Under the metric h, X becomes a complete non-compact Riemannian manifold with a corresponding Laplacian GAMMA(h). Consider solutions of the differential equation. (*) GAMMA(h)u + lambdaq(x)u = 0 onX where q(x) is a real analytic function on partial derivative X and lambda is-an-element-of C. Our main result is a representation theorem for all solutions of equation (*). The theorem is a generalization of a representation formula established by Helgason and Minemura for solutions of the Helmholtz equation on hyperbolic space.