Bethe Ansatz and Q-operator for the open ASEP

In this paper, we look at the asymmetric simple exclusion process with open boundaries with a current-counting deformation. We construct a two-parameter family of transfer matrices which commute with the deformed Markov matrix of the system. We show that these transfer matrices can be factorized into two commuting matrices with one parameter each, which can be identified with Baxter's Q-operator, and that for certain values of the product of those parameters, they decompose into a sum of two commuting matrices, one of which is the usual one-parameter transfer matrix for a given dimension of the auxiliary space. Using this, we find the T-Q equation for the open ASEP, and, through functional Bethe Ansatz techniques, we obtain an exact expression for the dominant eigenvalue of the deformed Markov matrix.