NIT Trichy: The reality of mystical experiences and reincarnation

The following is an excerpt from a samvaad (dialogue) session with Acharya Prashant.

Questioner (Q): Scientists have been trying to prove the existence of the soul for a long time now. They are doing research about near-death experiences, out-of-the-body experiences, and reincarnation. Eleven to fourteen percent of the average American population who are mentally and physically healthy have had an out-of-the-body experience at some point in their lives, as per the statement in the medical journal.

Dr. Ian Stevensen, a famous psychiatrist, did research on reincarnation and NDEs (near-death experiences). One of the research subjects, a lady who was a patient with a severe disease, stated that she was able to see herself and her body, like from an external camera. Dr. Ian Stevensen also says that phobias are related to the mode of death of the persons in their previous lives. For example, if somebody died in one of the previous lives by drowning, the person would be afraid of water, and there are other statements from the doctor supporting the claim on similar lines. Many Hindu scriptures like the Bhagavad Gita also say that the soul leaves one body and then gets a new body.

My question is, does reincarnation really happen, and what is the soul?