Difference analogue of the Treibich–Verdier operator

We will consider the problem of extending an one-dimensional finite-gap Schrödinger operator to a second-order difference operator that depends on a small parameter and commutes with some operator of odd order. It is assumed that if the small parameter tends to zero, the second-order difference operator will transform into a Schrödinger operator. We construct such an extension for the Treibich-Verdier finite-gap operator.