Here's a webpage on the 5^th Tennessee Regiment:

http://www.tngennet.org/civilwar/csainf/csa5.html

Note that during its organizational history more than one Company "H" will be found with this regiment. You should also be aware that James T. Harper also has a service file with Wilson's 21st Tennessee Cavalry.

Of course you can always order copies of this man's service files through the message board. It's also possible that the State of Arkansas has a pension application registered in his name.