well you have the security, that should cover one month, and you can always sue in small claims court for the rent lost advertising damages etc,over and above the deposit
BUT you have a duty to re-rent asap. You cannot sit on your butt for 3 months then claim he owes 3 months rent...
OR you cannot advertise the apartment for say $100 a month more then say to a judge, no one rented it at that price, the judge wont force a tenant to pay you for the months it was empty.
PS you got your security deposit at the closing....all leases and deposits get transferred at the closing, its in the sales price..