We discuss the errors introduced by level truncation in the study of boundary renormalization group (RG) flows using the truncated conformal space approach (TCSA). We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field φ (13) , we propose a RG equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the

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... nd we compare the predictions with data obtained using thermodynamic Bethe ansatz equations.