How can I hang a closet rod where there are no studs?
It is possible to fabricate some sort of support no matter what, it's more about what is the most expedient method of support that is not completely ugly. As mentioned by maple_shaft, there would typically be an intermediate stud between the corner and the window. It is conceivable that 24" spacing is being used and the only structure is at the window and at the corner. The window would typically have a least 3" width of rough framing at the side, mostly behind the trim, but allowing for frame thickness and rough-in allowance, there should be a bit of framing behind the wall finish you can nail into right along the edge of the trim. Even if completely behind the trim, you could, for instance, use a 1x6 support and dado it into the trim so that it has positive attachment to the underlying framing.

So now you have two points of support, one either side of the clothes pole. A 1x6 will be adequate to span between these points and look compatible with the trim. I agree you should use a proper clothes pole instead of copper. I assume it is the hard type that is somewhat rigid. (The soft type is obviously unsuitable) Even if it appears sturdy initially, under constant load, it will slowly bend over time (creep deformation) until it pulls out of it's support and falls.