The ordinary and matrix continued fractions in the theoretical analysis of Hermitian and relaxation operators

Abstract

We consider the theory of the resolvent for Hermitian or relaxation operators, and we address the problem of the explicit evaluation of the Green's function. The continued fractions are shown to be an efficient and natural calculational tool. With respect to the literature, we provide here for the first time a systematic theory, which develops directly from the general Dyson equation. Our novel treatment allows us to extend the theory of ordinary continued fractions from scalar to matrix parameters; it provides a unified formal treatment of both Hermitian and relaxation operators; it makes transparent the natural link, overlooked in the literature, between continued fraction approach and renormalization group techniques; finally it allows to establish the relationship with the moment method. A few examples are also reported to illustrate some relevant numerical or applicative aspects.