2001 SAAB 9-5 problems starting

My 2001 saab 9-5 runs well and starts without a glitch most of the time. Other times however, it will not start at all. At these times, the engine does not turn over, and I do not hear any clicking noises coming from the starter. This used to happen once every two or three weeks, but now it happens with increasing frequency, especially if I drive the car, turn off the engine, and then try to start the car again only a short time later. The car has a new battery. I was told this could be the alternator, but don't want to spend the money unless i'm sure that's what it is. The radio and the power windows and all of the electrical things work, even when the car is not starting. Do you have any idea what it might be? I would be very grateful for some good advice.