Most sleep centers are multi-disciplinary. Having said that, most physicians running sleep labs are pulmonologists, followed by neurologists, internists and psychiatrists. This is due to prior major focus on breathing related sleep disorders. This is changing with more attention being paid to insomnia and other similar sleep disorders.
Sleep centers will remain multi-disciplinary but less pulmonary people will be interested on pursuing an additional year of fellowship after the "grandfathering" period is over.
In addition, there is so much interface of sleep issues with psychiatry. There is already so much research on sleep symptoms and lab findings being considered potential biomarkers for psychiatric disorders. For example, insomnia is increasingly being thought of as an imporant predisposing factor rather than just one of the symptoms of depression. So keep tuned, more will come...