  In this paper we extend to the difference case the notion of
Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks
for the theory of deformations of Hamiltonian operators. A local scalar
difference Hamiltonian operator is a polynomial in the shift operator and its
inverse, with coefficients in the algebra of difference functions, endowing the
space of local functionals with the structure of a Lie algebra. Its
Poisson-Lichnerowicz cohomology carries the information about the center, the
symmetries and the admissible deformations of such algebra. The analogue notion
for the differential case has been widely investigated: the first and most
important result is the triviality of all but the lowest cohomology for first
order Hamiltonian differential operators, due to Getzler arXiv:math/0002164 .
We study the Poisson-Lichnerowicz cohomology for the operator $K_0 =
\mathcal{S} - \mathcal{S}^{-1}$, which is the normal form for $(-1,1)$ order
scalar difference Hamiltonian operators; we obtain the same result as Getzler
did, namely $H^p(K_0)=0$ $\forall p > 1$, and explicitly compute $H^0(K_0)$ and
$H^1(K_0)$. We then apply our main result to the classification of lower order
scalar Hamiltonian operators recently obtained by De Sole, Kac, Valeri and
Wakimoto arXiv:1806.05536
