Correlations in one dimensional quantum impurity problems with an external field or a temperature

We discuss in more details the theory of low energy excitations in quantum impurity problems with an external field at vanishing temperature, giving further support to results of the previous paper. We then extend these results to the next order at low frequency, obtaining in particular the exact expression, as a function of the bias, of the first two derivatives of the response function $\chi''(\omega)$ at $\omega=0$ in the double well problem of dissipative quantum mechanics. We also extend our approach to the case of non vanishing temperature and no external field. Fendley et al. had obtained in that case an expression for the Hall conductance with a single impurity using a Landauer-B\"uttiker type approach. We recover their result in the framework of linear response theory using renormalized form-factors. We also obtain, as a function of the temperature, the first derivative of the response function $\chi''(\omega)$ at $\omega=0$ in the double well problem of dissipative quantum mechanics.