Chevrolet Repair: corvette electrical, battery recharger, voltmeter

Question
crank car for short drive.  voltmeter trends toward down reading, then moves back to normal.  alternator checks good.
can shut engine down during low cycle and it will not restart.
last time battery recharger went to maximum at 10 amps reset and started again.  jump started car it fell below 0 and then came back up on voltmeter.  car then ran fine and restarted after coming home.  starter is new and problem is same as with old starter.  no fuses are being blown.  ideas?

Answer
Connections corroded?  How old is this car?  Battery could be bad.  After letting the car sit overnight, go out and take a reading.  Then turn on the key to run position, headlights on, and blower motor on; and watch the reading drift down.  How fast and how low.  Should not get lower than 12.2V.  

Try to start the car.  How low does the battery reading get?  If below 11.9V your battery is suspect.  If lower than 11.2V, get a new one.  If in the 10V range, battery is shot.  

Test the cables on the Ohm setting.  You should not see any more resistance than .03 ohms.  Between all connections in a single circuit, you should not have more than .1V voltage drop.