Schwinger and anomaly-free Kohn variational principles and a generalized Lanczos algorithm for nonsymmetric operators
Hans‐Dieter Meyer, J. Horáek, Lorenz S. Cederbaum · Physical Review A · 1991
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.