Car hesitates & shuts off. Check gauge & oil pressure lights then illuminate?

What is your question?

The problem normally happens if I have to make multiple stops which causes me to start and stop the car frequently. I know the problem is about to happen because when I apply the gas the car hesitates for about 2-3 seconds then vibrates; shuts off with the keys still in the start position and then finally the check gauge & oil pressure lights will come on. I know it has to do with something electrical or overheating because if I leave the car sitting for over an hour after this occurs, it starts right up and drives like nothing happens. when i take it to the shop they never find anything even though i explain when and how this happens. What could be the issue?