No, schism does not allow you to manifest powers when you are unconscious
Although divided minds created via Schism To act independently of each other, the "second mind" should always be affected by the conditions that apply to the character (with the explicit exception of haste effects). Nothing in the wording implies that one mind can be unconscious and the other not; if the character is unconscious, both minds are. If the character naturally falls asleep, both minds should sleep.
If affected by the sleeping spell, the schism would end; it's a constraint effect, and if you fail a backup against a constraint effect, Schism will end automatically one way or another, so you can not use To sleep induce a state of sleep and keep SchismThe advantages of.
So, how can you manifest powers by being unconscious?
The only way to protest while I was unconscious, I could find in the Psionics Unleashed or Psionics Expanded: Advanced Psionics Guide handbooks is the Lucid Dreaming trait, which allows you to access your powers while dreaming:
… you know when you dream and have access to your pool of power points and your powers known in the dream.
The disadvantage is that the points spent while dreaming are not recovered when you wake up normally, so that one of the Supervised sleepThe advantages of.
Throw the net a little wider (Dreamscarred produced a lot splatbooks based on psionics), Psionics Augmented: exploits and powers adds unconscious Inconscient exploit, providing the following benefit:
The first round after you have lost consciousness, you can use the unconditional power exploit to manifest power as if you were stunned.
I would say that allowing a character to use the exploit to manifest power just after falling asleep. To do this, you must at least be a level 13 multiplier, since the use of unconditional power supply increases the energy cost of 8, so the total cost of the event Supervised sleep In this way, it would give 13 points and you have to invest two feats.