Subsidiary minimum principles for scattering parameters

Abstract

We denote as a minimum one in which a quantity $B$ of physical interest is represented as the minimum value with respect to variations in a trial function ${\ensuremath{\psi}}_{t}$ of a functional $F({\ensuremath{\psi}}_{t})$; $F$ 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 ${\ensuremath{\psi}}_{t}$, 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 ${\ensuremath{\psi}}_{t}$. We show here that for a wide class of problems for which primary minimum do not exist, useful and rigorous secondary or minimum principles are available. That is, we construct a functional ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ whose minimum value is reached for ${\ensuremath{\psi}}_{t}$ equal to some function $\ensuremath{\psi}$ 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 $B=B(\ensuremath{\psi})$, then a study of ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ provides a powerful tool for the estimation of $\ensuremath{\psi}$ and therefore $B$, though $B({\ensuremath{\psi}}_{t})$ is not normally a variational bound on $B(\ensuremath{\psi})$. Subsidiary minimum have recently been obtained for the approximation of the auxiliary functions that appear in the variational principle for the matrix element (${\ensuremath{\chi}}_{n}$, $W{\ensuremath{\chi}}_{m}$), where ${\ensuremath{\chi}}_{n}$ and ${\ensuremath{\chi}}_{m}$ 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 $g(\ensuremath{\epsilon})$ of a bound system with $\ensuremath{\epsilon}$ 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.