Is your nice battery a "Sealed Maintenance Free - Lead Acid Battery" ? or is it a Lantern Battery ? or is it any other kind of battery?
What's a PS2 Power supply? did you mean Sony Playstation 2 ? or IBM Personal System 2 ?
If it's a lead acid battery then you cannot use it till it's empty as it will damage the battery.
You will need circuitry designed to charge a lead-acid battery, with low battery voltage detection.
An empty battery isn't much use, is it?
Holy crap. 199$ for that old crap battery? I buy them for 7$ here.
NO, don't charge it with a Playstation power brick. Use a dedicated 6V lead acid battery charger wall wart which can detect the battery's state of charge. Look for the one with good reviews and buy.
Try, Aliexpress or Banggood.
what can do for a 12 v 7 Ah battery charger with arduino? is it is safer to do with MOSFET with pulse modulation method? any suggestions?
I'll add to the pile. If you discharge a lead acid battery until it is empty, you will damage it.
Terminal charge is 2.35V per cell.
I strongly suggest you find an IC designed to charge a lead acid battery. Although you could also just get a step-down module, connect it to to the power supply, and set it to a max of 3 x 2.35 = 7.05V and 0.5A maximum current.
There are many simple circuits on line for lead acid battery chargers.
Google "lead acid battery charger circuit" for examples like this one.