Remarks on Scattering Theory

Some consequences of the construction of potentials from phase shifts are derived. The question of selecting a unique potential by asymptotic requirements, and the connection with analyticity of the $S$-matrix is discussed. An expression for the variational derivative of the potential with respect to the phase shift is derived. A simple exact relation between the value of the potential at the origin and the first moment of the phase shift emerges as a result. The variational derivative is made the basis of a "phase-shift perturbation method." One "unperturbed" $S$-matrix suggests itself naturally, and is here called the "intrinsic" $S$-matrix of the bound state. The "intrinsic" potential is discussed. Variational derivatives of one phase shift with respect to another, as well as an extension of the phase-shift perturbation method near the intrinsic $S$-matrix to the presence of a tensor force, are given.