Today I was poking around the interwebs looking to find any type of medical explanation for Ms. Sagee’s condition. I found this on Wikipedia and quoted it below. However, this does not explain how her students were able to see (and feel) her double.
The syndrome of subjective doubles is a rare delusional misidentification syndrome in which a person experiences the delusion that he or she has a double or Doppelgänger with the same appearance, but usually with different character traits and leading a life of its own. Sometimes the patient has the idea that there is more than one double. The syndrome is usually the result of a neurological disorder, mental disorder or some form of brain damage, particularly to the right cerebral hemisphere. In some cases, the syndrome (also referred to as syndrome of Christodoulou) may be the result of delusional elaboration of autoscopic phenomena.
Sometimes the delusion takes the form of a conviction that whole or part of the patient’s personality has been transferred into another person. In this case depersonalization may be a symptom. One example from medical literature is of a man who became depersonalized after an operation and was convinced his brain had been placed into someone else’s head. He later claimed he recognized this person.