Memory function Lanczos method

Once a tridiagonal representation of M is obtained, one can use standard methods to diagonalize the tridiagonal matrix and obtain the eigenvalues (in this particular aspect, the Lanczos algorithm, as commonly used in the literature, differs from the other memory function methods where the Green s... 

The advantages of this kind of formulation stand out not only in terms of elegance and beauty (the moment method, the Lanczos method, and the recursion method are relevant but particular cases of the memory function equations), but also in the possibility of providing insight into a number of problems, such as the asymptotic behavior of continued fraction parameters and their relationship with moments, the possible inclusion of nonlinear effects, the introduction of the concept of random forces, and so on. 

To verify the equivalence of the memory function approach to the recursion method or the Lanczos method, it is sufficient to note that the state l/ i+i) defined via Eq. (3.39d) coincides with the state... 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

In this chapter, we also discussed several schemes that allow for the computation of scalar observables without explicit construction and storage of the eigenvectors. This is important not only numerically for minimizing the core memory requirement but also conceptually because such a strategy is reminiscent of the experimental measurement, which almost never measures the wave function explicitly. Both the Lanczos and the Chebyshev recursion-based methods for this purpose have been developed and applied to both bound-state and scattering problems by various groups. 

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