Can bargaining be done with the convicted as shown in Law Abiding Citizen?

Well it so stands that you are wrong (and I don't know how to mark you!). On a more serious note I'll explain where you are mistaken.

If we can not prove for sure that a convicted is guilty certainly, the
prosecutor can bargain with the prisoner to relieve him from the case
even before the court gives its verdict!

That's not entirely correct. If a prosecutor doesn't have enough evidence to prove the defendant guilty then the defendant walks free, but it is still decided at court (or the case can be dismissed altogether by the judge).

The concept you are missing out on is Plea Bargaining; whereby the defendant agrees to plead guilty to a particular charge in return for some concession from the prosecutor.

In the movie, Rice makes a deal with Darby that in return for pleading guilty to third-degree murder, he will provide testimony that will send Ames to death row for what is, essentially, a robbery charge. So the defendant (Darby) makes a deal with the prosecutor (Rice) to plead guilty for a reduced charge.