Smooth Feshbach map and operator-theoretic renormalization group methods.
Having been created by Feshbach in the 1950s as a tool of perturbation theory in Hilbert space, three of the present article's authors have, in recent years, developed the Feshbach projection method to a tool of spectral analysis of perturbed operators. Given an operator \(T\) with known spectrum, the method concerns perturbations \(H:=T+W\) of \(T\) to which an operator \(F(H,W)\) can be assigned whose main spectral properties coincide with those of \(T\) at \(0\). Since \(F\) may be considerably simpler to analyze (it may even be a finite matrix), this fact has turned out to be, apart from its mathematical appeal, of particular use in applications. The construction of every particular Feshbach map \(F_P\) is based, in the earlier work, on an appropriately chosen ``cutoff'' projection \(P\). The elements of the domain of \(F_P\) are called the Feshbach pairs associated with \(P\). The present paper generalizes the notion of Feshbach map by replacing the projection \(P\) by an arbitrary positive operator \(\chi\leq1\). The scope of the analysis are such choices of \(\chi\) that make \(F_\chi\) ``smooth'' in the sense that a new operator-theoretic renormalization group method can be established with the help of \(F_\chi\). The method is illustrated with the example of a class of Fock space Hamiltonians.
scientific article; zbMATH DE number 1992838