Hello,
There are many causes of sunken
under eyes,the common ones being dehydration,
eye conditions like nanophthalmos,microphthalmos,malnutrition,anorexia nervosa,protein losing enteropathy,underactive thyroid gland and lacrimal tumors.
I would suggest you to consult an ophthalmologist and discuss these with him.
Hope it helps.Take care and pls do keep me posted on how you are doing or if you have any additional doubts.Kind regards.