  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 it 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.
