Generalized Caroli formula for the transmission coefficient with lead-lead coupling

We present a generalized transmission coefficient formula for the lead-junction-lead system, in which interaction between the leads has been taken into account. Based on this formula, the Caroli formula could be easily recovered and a transmission coefficient formula for interface problem in the ballistic system can be obtained. The condition of validity for the formula is carefully explored. We mainly focus on heat transport. However, the corresponding electrical transport could be similarly dealt with. Also, an illustrative example is given to clarify the precise meaning of the quantities used in the formula, such as the concept of the reduced interacting matrix in different situations. In addition, an explicit transmission coefficient formula for a general one-dimensional interface setup is obtained based on the derived interface formula.