Completeness of the natural modes for quantum mechanical potential scattering

The set of natural modes, associated with quantum mechanical scattering from a central potential of finite-range is shown to be complete. The natural modes satisfy a non-Hermitian homogeneous integral equation, or alternatively, are solutions of the time independent Schroedinger equation subject to a recently formulated nonlocal boundary condition (the quantum mechanical extinction theorem). An expansion theorem similar to that of Hilbert--Schmidt is formulated, valid for values of the solution of the scattering integral equation inside the range of the potential. The boundary conditions generated by the quantum mechanical extinction theorem are shown to be closely connected with the Jost function