Subsidiary minimum principles for scattering parameters

We denote as a "primary minimum principle" one in which a quantity $B$ of physical interest is represented as the minimum value with respect to variations in a trial function ${\psi }_t$ of a functional $F\left({\psi }_t\right)$; $F$ then provides a variational upper bound on $B$. (The Rayleigh-Ritz principle for the ground-state energy of a system is a familiar example.) If $F$ is quadratic in ${\psi }_t$, the variational property of $F$ enables one to determine the linear parameters relatively easily, but the minimum property is required if the nonlinear parameters are to be determined in a way which allows for systematic improvement of ${\psi }_t$. We show here that for a wide class of problems for which primary minimum principles do not exist, useful and rigorous secondary or "subsidiary minimum principles" are available. That is, we construct a functional $F^{\prime }\left({\psi }_t\right)$ whose minimum value is reached for ${\psi }_t$ equal to some function $\psi$ of dynamical interest. (The Rayleigh-Ritz method provides a subsidiary minimum principle for the approximate determination of the ground-state wave function of a system.) If $B=B\left(\psi \right)$, then a study of $F^{\prime }\left({\psi }_t\right)$ provides a powerful tool for the estimation of $\psi$ and therefore $B$, though $B\left({\psi }_t\right)$ is not normally a variational bound on $B\left(\psi \right)$. Subsidiary minimum principles have recently been obtained for the approximation of the auxiliary functions that appear in the variational principle for the matrix element (${\chi }_n$, $W{\chi }_m$), where ${\chi }_n$ and ${\chi }_m$ are bound-state wave functions and $W$ is an arbitrary operator. Here we extend the method to the estimation of matrix elements of the Green's function $g\left(\epsilon \right)$ of a bound system with $\epsilon$ below the continuum threshold energy. The response of the system to an external perturbation can be represented by matrix elements of this type. While no new results on the bound-state problem are obtained, our formulation is a convenient starting point for the further extension of the method to continuum problems. The new result obtained here is the derivation of a subsidiary minimum principle for the problem of scattering of a projectile by a target whose bound-state wave function is only imprecisely known. The subsidiary minimum principle allows for systematic improvement of the closed-channel component of the trial scattering wave function that appears in a Kohn-type variational calculation of the scattering amplitude.