A Representation Theorem for Solutions of the Helmholtz Equation and Resolvent Estimates for The Laplacian

Publisher Summary This chapter presents a representation theorem for solutions of the Helmholtz equation. The representation of solutions by the generalized eigenfunctions has some relation to the spectral theory of the Laplacian on Rn . It is this relation that makes the theorem interesting. A similar representation theorem by generalized eigenfunctions can be established for solutions on hyperbolic n-space, Δ denoting the hyperbolic Laplacian. The chapter also presents some results related to questions of analytic continuation of G(k)f into the lower half-plane.