Rank two bundles on P^n with isolated cohomology

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair of values $(k,n-k)$. We show that on $\mathbb P^8,$ if $H^3_*(\mathcal E)=H^4_*(\mathcal E)=0$, then $\mathcal E$ must be decomposable. More generally, we show that for $n\geq 4k$, there is no indecomposable bundle $\mathcal E$ for which all intermediate cohomology modules except for $H^1_*, H^k_*, H^{n-k}_*, H^{n-1}_*$ are zero.