This paper explores the relationship between scattering and bound waveforms in gravitational two-body dynamics using the Schwinger-Dyson equations and Bethe-Salpeter recursion. The authors show that the tree-level scattering waveform can be analytically continued to the bound waveform, which is confirmed by an independent calculation in the Post-Newtonian (PN) expansion. They demonstrate consistency with the Damour-Deruelle prescription for orbital elements in the quasi-Keplerian parametrization and with the analytic continuation of radiated energy and angular momentum at 3PM order. The paper begins by introducing the setup and notation for gravitational two-body dynamics, focusing on the effective field theory description of point particles. It then discusses the derivation of classical wavefunctions from scattering to bound states, using the linearized Schwarzschild metric as a toy example. The authors show that the single branch cut prescription for analytic continuation relates scattering and bound wavefunctions, with the bound state wavefunction obtained by taking the residue of the bound state pole after analytic continuation. The paper then explores the scattering and bound matrix elements from the Schwinger-Dyson equations, showing how the conservative two-massive-particle-irreducible (2MPI) kernel can be used to perform the analytic continuation. The authors also discuss the analytic continuation of the Post-Minkowskian (PM) waveform, showing that it can be extended to the bound counterpart in terms of binding energy. They test this conjecture against the direct calculation of PN multipoles using the quasi-Keplerian parametrization and show how the scattering-to-bound map can be derived by studying the integration over the retarded time of the fluxes. The paper concludes by summarizing the current status of relations between scattering and bound observables, emphasizing the universality of classical particle dynamics and the importance of the analytic continuation in connecting scattering and bound waveforms. The authors also discuss the implications of their results for the development of gravitational wave templates and the connection between scattering and bound dynamics in general relativity.This paper explores the relationship between scattering and bound waveforms in gravitational two-body dynamics using the Schwinger-Dyson equations and Bethe-Salpeter recursion. The authors show that the tree-level scattering waveform can be analytically continued to the bound waveform, which is confirmed by an independent calculation in the Post-Newtonian (PN) expansion. They demonstrate consistency with the Damour-Deruelle prescription for orbital elements in the quasi-Keplerian parametrization and with the analytic continuation of radiated energy and angular momentum at 3PM order. The paper begins by introducing the setup and notation for gravitational two-body dynamics, focusing on the effective field theory description of point particles. It then discusses the derivation of classical wavefunctions from scattering to bound states, using the linearized Schwarzschild metric as a toy example. The authors show that the single branch cut prescription for analytic continuation relates scattering and bound wavefunctions, with the bound state wavefunction obtained by taking the residue of the bound state pole after analytic continuation. The paper then explores the scattering and bound matrix elements from the Schwinger-Dyson equations, showing how the conservative two-massive-particle-irreducible (2MPI) kernel can be used to perform the analytic continuation. The authors also discuss the analytic continuation of the Post-Minkowskian (PM) waveform, showing that it can be extended to the bound counterpart in terms of binding energy. They test this conjecture against the direct calculation of PN multipoles using the quasi-Keplerian parametrization and show how the scattering-to-bound map can be derived by studying the integration over the retarded time of the fluxes. The paper concludes by summarizing the current status of relations between scattering and bound observables, emphasizing the universality of classical particle dynamics and the importance of the analytic continuation in connecting scattering and bound waveforms. The authors also discuss the implications of their results for the development of gravitational wave templates and the connection between scattering and bound dynamics in general relativity.