Unstable Lazarsfeld-Mukai bundles of rank 2 on a certain K3 surface of Picard number 2

Let $g$ and $c$ be any integers satisfying $g\ge 3$ and $0\leq c\leq \lfloor\frac{g-1}{2}\rfloor$. It is known that there exists a polarized K3 surface $(X,H)$ such that $X$ is a K3 surface of Picard number 2, and $H$ is a very ample line bundle on $X$ of sectional genus $g$ and Clifford index $c$, by Johnsen and Knutsen([J-K] and [Kn]). In this paper, we give a necessary and sufficient condition for a Lazarsfeld-Mukai bundles of rank 2 associated with a smooth curve $C$ belonging to the linear system $\mathrm{\mid }H\mathrm{\mid }$ and a base point free pencil on $C$ not to be $H$-slope stable.