First it should be noted that your arrow can only have the energy you put into it. The energy you put into it is force * distance as you draw it back. More specifically, it is the area under the force-draw curve:
This curve plots the amount of force you have to apply as a function of the distance you draw, for a few bows. For instance, according to this chart, if you want to draw the 125lb flight bow back 15 inches, you will need to apply 60lb of force. The area under the curve is the maximum amount of energy that can go into your arrow. If the force is higher, or if the draw distance is longer, then you will get a faster arrow.
In practice, not all of the energy of your draw goes into the arrow. Most modern bows are about 80% efficient, meaning 20% of the energy goes into moving the bow or the string, or is dissipated as noise and heat.
Your magic bow could do better than a modern bow, while still respecting physics, by increasing that efficiency. So it could at most put in 25% more energy than a standard bow - assuming it has the same draw weight and the same force-draw curve as the standard bow.
We can imagine that your magic bowstring is perfectly efficient - every bit of energy you put in comes out. To attain almost 100% efficiency it is necessary only that the bow doesn't significantly move, which seems feasible with a firm hand.
There is one other thing your magic bow could do. Compound bows have a different draw curve. The force increases rapidly to a set level, and remains at that level for the rest of the draw, until the very end of the draw when it eases off a little so you can hold it there. The force-draw curve for a compound bow looks like this:
This is useful because the force the user needs to apply is consistently at the maximum force they can draw, instead of starting off easy and getting harder only at the end. That increases the area under the curve, and therefore the energy of the arrow.
If your magic bow had a draw-force curve that looks like that, it could put about twice as much energy into the arrow as a non-compound bow. (Most fantasy bows are non-compound bows; compound bows are the complicated looking ones with pulleys.)