Luttinger liquid with one impurity: equivalent field theory and duality

We derive by functional integral method 1-dimensional theory equivalent to Luttinger liquid with one impurity. It is shown that the single quantity has to be calculated for the formulation the effective field theory is the electron density jump at the impurity. We show that duality which relates models with electron-electron attraction and repulsion, and simultaneous exchange of transition and reflection coefficients, is an exact property of the theory.