Tomonaga-Luttinger liquid with reservoirs in a multi-terminal geometry

We propose a formalism which uses boundary conditions imposed on the Luttinger liquid (LL) to describe the transport properties of a LL coupled to reservoirs. The various boundary conditions completely determine linear transport in the joint system reservoirs+LL. As an illustration we consider an exactly solvable microscopic model in a multi-terminal geometry for which such boundary conditions can be explicitly derived; in this model the Landauer-B\"uttiker formalism fails: if it were valid, the relation between the conductance matrix elements and the reflection and transmission coefficients could yield negative probabilities. We then apply our formalism to a discussion of shot noise through an impurity in a LL connected to two reservoirs.