Vector bundles trivial on complement of a hypersurface section

Question

Assume we have a smooth projective variety $X$ over a field (finite field if matters). Consider a smooth hypersurface section $H$ of $X$. Let $Spec(A)=U=X \setminus H$ be an affine variety. I want to know whether there are any nice description of vector bundles on $X$ such that they are trivial on $U$ and $H$? My suspicion was that they are all direct sums of line bundles of the form $O(nH)$ is that correct?

Edit: Let's assume that we are talking about rank $m$ vector bundles that are subsheaves of the constant sheaf given by the vector space $F(X)^{\oplus m}$ which $F(X)$ is the function field or equivalently field of fractions of $A$. Does that change anything?

Answer 1

In many cases, $\mathcal{O}_X(n \cdot H)$ itself will not be trivial on both $H$ and $U$, because it will be nontrivial on $H$. For example, if $H \subset \mathbb{P}^n$ is a degree $d$ hypersurface, then $ \mathcal{O}_{\mathbb{P}^n}(H) \simeq \mathcal{O}_{\mathbb{P}^n}(d)$ and so $\mathcal{O}_{\mathbb{P}^n}(H)|_H \simeq \mathcal{O}_H(d)$ which is non-trivial as long as $n> 1$, for instance because it is ample.

EDIT: I realize what this shows is: not all sums of line bundles of the form $\cal{O}_X(n \cdot H)$ are trivial on both $U$ and $H$. All the same it seemed relevant.