Schwinger and anomaly-free Kohn variational principles and a generalized Lanczos algorithm for nonsymmetric operators.

Abstract

The Schwinger variational principle is combined with the Lanczos algorithm. It is shown that the straightforward approach where the Lanczos basis is generated by V-${\mathit{VG}}_{0}$V is not very useful. The difficulties arise because of the unboundedness of the Schwinger operator V-${\mathit{VG}}_{0}$V for unbound potentials (such as, e.g., the Yukawa potential). A modification of the Schwinger operator is presented that resolves the problem. This approach leads to a new Lanczos recursion for nonsymmetric (and non-Hermitian) operators yielding nonorthogonal basis functions. The resulting approach is shown to be equivalent to the continued-fraction method of Hor\'a\ifmmode \check{c}\else \v{c}\fi{}ek and Sasakawa. To illuminate the particular properties of the Schwinger-Lanczos basis we have applied it as well to other variational principles like the C\ifmmode \tilde{}\else \~{}\fi{} functional, the Newton variational principle, and the Kohn variational principle. To treat the multichannel problem we have adopted a modified band-Lanczos algorithm. This allows for a more efficient computation of the off-diagonal T-matrix elements than previous approaches.