Miguel Ballesteros

Abstract

The spectral renormalization method is a widely used mathematical tool in spectral theory, particularly in low-energy quantum field theory. Its original introduction marked a milestone in the field. Although inspired by physics and commonly referred to as the renormalization group, this method is neither a group nor a semigroup (more accurately, it is a flow). It was not until 2015 that a genuine flow (or semigroup) structure was introduced, using an innovative definition of the renormalization of spectral parameters. However, that spectral renormalization flow is not compatible with the smooth Feshbach–Schur map—a point acknowledged as an open problem in the literature. This incompatibility is a notable drawback, since smoothness of the Feshbach–Schur map is a key property that greatly simplifies proofs and makes it the preferred tool in most of the literature. In the present paper, we resolve this open problem by constructing a spectral renormalization flow that is based on the smooth Feshbach–Schur map.