Why Recur?

Abstract

The recursion method is having an ever widening development and range of applications, and in opening this conference I want to indicate what I think are some of the ideas behind this. I need hardly explain to this audience what the recursion method is, beyond defining my notation. Let me write the recursion relation as (see p.70: references purely in the form of page numbers refer to the review articles [1], [2], [3])

$${b_{{n + 1}}}{u_{{n + 1}}}{\text{ = }}H{u_{n}} - {a_{n}}{u_{n}} - {b_{n}}{u_{{n - 1}}}$$

((1))

where H is the Hamiltonian or other operator in some matrix form, the u_n constitute the set of new basis functions in column vector form, and the a_n,b_n are the coefficients. The coefficients are used to write a continued fraction to represent the matrix element of the Greenian or resolvent

$${G_{00}}(E) = < {u_0}\left| {{{(E - H)}^{ - 1}}} \right|{u_0} >.$$

((2))