Title: Spectral properties of the renormalization group
The renormalization group (RG) approach is largely responsible for the
considerable success which has been achieved in developing a
quantitative theory of phase transitions. Physical properties emerge
from considering the spectral properties of the linearization of the
RG map at a fixed point. We consider real-space RG for classical
lattice systems. The linearization acts on an infinite-dimensional
Banach space of interactions. At a trivial fixed point (zero
interaction), the spectral properties of the RG linearization can be
worked out explicitly, without any approximation. Current results are
for the RG maps corresponding to decimation and majority rule. They
indicate spectrum of an unusual kind: dense point spectrum for which
the adjoint operators have no point spectrum at all, but only residual
spectrum. This may serve as a lesson in what one might expect in more
general situations.