Causality and Dispersion Relations

according analytic continuation analytic function analytic properties analyticity domain angular momentum apply assumption asymptotic behavior bound causality condition compact convergence corresponding cutoff potential defined derived differential discussion dispersion relation energy entire function exponential finite number fixed momentum transfer follows Fourier transform given Green's function half-plane holomorphic implies incident wave packet inequality infinity integral equation interaction Jost k-plane Let us consider locally integrable function Mandelstam meromorphic number of poles oscillator outgoing wave partial-wave expansion particle Phys positive imaginary axis propagation quadrant quantum radius real axis Regge poles region resonance right-hand side S-function S-matrix s-wave satisfies scattered wave scattering amplitude Schrödinger Schrödinger equation shown singularities ſº solution space sphere square integrable Substituting supp temperate distribution term test functions theorem theory total scattering amplitude unitarity unitarity condition vanishes variables wave function wave packet York Yukawa-type potentials zeros