Improvement of Rayleigh–Ritz Eigenfunctions
Walter Kohn · SIAM Review · 1972
A common method of obtaining approximate eigenfunctions of a linear boundary value problem, $H\psi = E\psi $ and homogeneous linear boundary conditions, is the Rayleigh–Ritz method. We consider here the case where $\psi $ is expanded as $\psi = \sum _1^N {c_n w_n } $ and the $w_n $ are solutions of a “standard” problem, with the same boundary conditions. The eigenvalue E and coefficients $c_n $ are conventionally determined from the condition $\delta [(\psi ,H\psi ) - E(\psi ,\psi )] = 0$. We show here (without rigorous proofs) that, for large N, most of the error of the $ u$th Rayleigh–Ritz function $\psi _ u $ at the point $x_1 $ is removed by the addition of the correction term $\Delta \psi _ u (x_1 ) \equiv (\psi _ u (x),(H - E_ u )g(x,x_1 ))$. Here $E_ u $ is the approximate Rayleigh–Ritz eigenvalue and $g(x,x_1 )$ is a function, largely arbitrary, except for a specified singular behavior at $x = x_1 $ (an appropriate kink in one dimension, a $[4\pi ({\bf r} - {\bf r}_1 )]^{ - 1} $ singularity in three dimensions, etc.). Examples are given and possible extensions are discussed.