Current tenant defaming me to potential new tenants

[Teresa Dinning]

We are currently moving out two tenants.  The cops have been called twice due to domestic violence.  The smaller of the two women moved out, but we've been told by the downstairs tenant that she's come back.  The larger woman, who said she was moving out voluntarily at the end of the month (they're paid up til then), was the aggressor. There's something wrong with her; she lies about things and is constantly trying to pick fights.  We're glad they're leaving.  

The downstairs tenant, who is also in the process of moving out, reported that the problem tenant is telling prospective tenants who stop because of the "for rent" sign that they shouldn't talk to me because I lie about everything.

Do I have any recourse? I can't prove anything unless I catch her in the act, which of course she won't do if she sees me nearby.  If they decide not to move out, I can move forward with a 30 day eviction due to lease violations (the cops being called to the unit would violate the "peaceable use" clause).  If they fail to pay on the first, I can start with a  3 day eviction for non-payment.  Until then, she could turn away dozens of potential future tenants.