Splitting DC Supply Rail with High Output Currents
If you could do what you want, you would win the Nobel Prize. Your supply puts out 19 volts at 500 ma, which is a power level of 9.5 watts. If you could use that to produce +/- 12 volts at 3 amps, a single side would produce 12 volts at 3 amps, or 36 watts. So you could then use 9.5 of those watts to feed the input of your converter, and have 26.5 watts for free. Of course, you could then cascade converters to run your house, run an electric car, or anything else a perpetual motion machine is good for.

As you might guess, this is not going to happen.