Unitary transformations are an essential tool for the theoretical understanding of many systems by mapping them to simpler effective models. A systematically controlled variant to perform such a mapping is a perturbative continuous unitary transforma
tion (pCUT) among others. So far, this approach required an equidistant unperturbed spectrum. Here, we pursue two goals: First, we extend its applicability to non-equidistant spectra with the particular focus on an efficient derivation of the differential flow equations, which define the enhanced perturbative continuous unitary transformation (epCUT). Second, we show that the numerical integration of the flow equations yields a robust scheme to extract data from the epCUT. The method is illustrated by the perturbation of the harmonic oscillator with a quartic term and of the two-leg spin ladders in the strong-rung-coupling limit for uniform and alternating rung couplings. The latter case provides an example of perturbation around a non-equidistant spectrum.
Effects of truncation in self-similar continuous unitary transformations (S-CUT) are estimated rigorously. We find a formal description via an inhomogeneous flow equation. In this way, we are able to quantify truncation errors within the framework of
the S-CUT and obtain rigorous error bounds for the ground state energy and the highest excited level. These bounds can be lowered exploiting symmetries of the Hamiltonian. We illustrate our approach with results for a toy model of two interacting hard-core bosons and the dimerized S=1/2 Heisenberg chain.
