With all due respect, I'm going to disagree with Jacek and the Baron and say, no, the archer does not provoke attacks of opportunity with every shot of a full attack action.
Last night I took a good look at the AoO rules on pp. 178-9 as well as the list and note on pg. 182 that Jacek referred to.
While there certainly seems to be some room for differing interpretations*, the sense I got was that an AoO is triggered by the type of action being attempted, not the elements of that action (whatever they may be).
Thus, IMHO, it is the taking of a ranged attack action (whether it is standard or full doesn't matter) that triggers a single AoO from adjacent, threatening foes. Thus, the number of arrows loosed in no way determines how many attacks of opportunity a foe gets should that foe have Combat Reflexes.
Circumstantially, this is backed up by the rule for attacks of opportunity provoked by moving through a foe's threat range: it doesn't matter how many squares you pass through within that threat range, your foe only gets a single AoO.
*in other words, as with any rule in the game, do what seems best to you as GM