I believe this is regulated by the Department of Transportation. They are working on new rules to be out for public comment within the year and to be set within three years which may limit which animals are permitted.
Current regulations require that the airline accommodate reasonable requests to fly with an ESA or PSD if the airline is contacted at least 48 hours prior to the scheduled departure time and the following described letter is offered on request.
These requirements for the letter are excerpted from
http://airconsumer.ost.dot.gov/rules/20030509.pdf
The letter:
* must not more than one year old
* must be on the professional's letterhead
* must be from a mental health professional
and must state all of the following:
(note: bolded phrases are of specific wording that should be used as presented and not paraphrased to minimize rejection of the letter)
1. That the passenger has a mental health-related DISABILITY that appears in the DSM-IV. Note it is not just a mental illness diagnosis, but a mental illness which*substantially limits one or more major life activities. Airlines are not permitted to require the documentation to specify the type of mental health disability or the specific diagnosis, but the letter must state that the diagnosis appears in the DSM-IV.
2. That the presence of the animal is*necessary*to the passenger's health or treatment during the flight and/or at the destination.
3. That the individual writing the letter is a licensed mental health professional and that the passenger is under his or her care. The individual writing the letter should clearly indicate what type of mental health care professional they are (psychiatrist, psychologist, clinical social worker, et cetera).
NOTE: Airlines may also require documentation including the date, type, and state of the mental health professional's license so it may be wise to just have them include that from the start so you don't have to go back and ask for it later.
Some airlines do call the professional's office to verify the contents of the letter.