What can I do about a mentally ill neighbour who stares all the time?

Are you sure that he's been diagnosed as mentally ill? It sounds like you're putting that label on him yourself. You have it backwards by saying that he knows his surroundings because he's psychotic - a person experiencing psychosis does NOT know their surroundings, not to mention that's something only a doctor can diagnose. If he does have a mental health impairment, maybe it's a disability that would explain him having to live with his parents and lacking social skills. There's really not much that can be done.