Postscript:

Since $s \equiv 2$ and $\Dslash^2 = \Delta^S + \half$, we also get

$$\spec(\Delta^S) = \set{(l + \half)^2 - \half = l^2 + l - \quarter : l \in \bN + \half}$$

with multiplicities $2(2l + 1)$ in each case. Note that

$$\spec(\Dslash^2) = \set{(l + \half)^2 = l^2 + l + \quarter : l \in \bN + \half}.$$

The operator $\ul{C}$ given by $\ul{C} := \Delta^S + \quarter = \Dslash - \quarter$ has spectrum

$$\spec(\ul{C}) = \set{l(l + 1) : l \in \bN + \half},$$

with multiplicities $2(2l + 1)$ again. This $\ul{C}$ comes from the Casimir element in the centre of $\sU(\gsu(2))$, represented on $\sH = L^2(\bS^2, S)$ via the rotation action of $SU(2)$ on the sphere $\bS^2$. There is a general result for compact symmetric spaces $M = G/K$ with a $G$-invariant spin structure, namely that $\Dslash = \ul{C}_G + \frac{1}{8} s$, or $\Delta^S = \ul{C}_G - \frac{1}{8}s$. This is a nice companion result, albeit only for homogeneous spaces, to the Schrödinger-Lichnerowicz formula. Details are given in Section 3.5 of Friedrich's book.