I have to admit that I don't know very much about schizophrenia. But from what I heard, I can say that people suffering from this disease are not frequently violent towards others. They tend to be aggressive towards themselves, hearing voices that tell them to commit suicide.
They might be afraid of people who wear blue jeans or think that wearing a certain hat will keep them save from rays from outer space. Something like that.
Of course we can't be sure how a superpowered man would react if he felt threatened by people wearing jeans, because he thinks that they are watching his every step. Maybe he would be aggressive, knowing that those people can't overpower him. But on the other hand there are so many people with jeans that he can't possibly attack them all. And in his troubled mind I guess he would assume that the remaining people in blue jeans will tell some superior might about his actions.
Superpowers or not, I think that a schizophrenic son of Superman would probably behave like most other schizophrenics and would rather try to kill himself than others. I might be wrong. I'm sure Beethoven and Nancy know more about the disease than I do.
