Crossfire 2005 won't start at times

We have a 2005 Crossfire. The cars is great except at times it will not start. The problem has no schedule and does not happen often, although it does appear to occur only in the afternoon. There seems to be no coorelation to a hot or cold engine. The last two times the car has failed to start it was after it had sat in the parking lot all day, parked at 8 AM attempted to restart at 4:30 PM (ambient temperature of 80° to 90°F). In these two cases after some time had passed, 3 + hours, the car started. Times before, it had failed during “shopping” trips (in and out of stores). Once it failed while being driven in stop and go traffic. We’ve had this car at the Chrysler dealer many times. They only know what the computer tells them and they say they cannot find the fault in the cars’ computer record. When we attempt to start the car, all the indicator lights illuminate but there is absolutely no signal sent to the starter when the key is turned. Once I observed the LED lights at the bottom of the instrument cluster did not illuminate when the door was opened. We are getting very frustrated. Any thoughts?

The next time the engine won’t start, turn the ignition switch on so the dash lights come on. Step on the brake pedal and place the transmission into neutral and then try starting the engine. If the engine starts there’s a problem with the park/neutral safety switch.

Tester

You may have an immobilizer malfunction. When the vehicle fails to start check the odometer display for a “start/error” message.