Baxter $Q$-operators for the integrable discrete self-trapping chain

For the integrable discrete self-trapping chain, we construct Baxter $Q$-operators as the traces of the monodromy of certain $M$-operators that act in the quantum and auxiliary spaces. With this procedure, we obtain two basic $M$-operators and derive some functional relations between them such as intertwining relations and Wronskian-type relations between two basic $Q$-operators.