List search with unique permissions

I'm the sharepoint owner and setting up a site, and I'm having an issue where I break inheritance.  If i have a list that has unique permissions, that list no longer shows up in search.  This specific list, i only want the owners group to be able to view.  I want the owners group to be able to search this list, but nothing returns in the search for this list.  As soon as I remove the unique permissions and go back to inheriting from the parent, i can search the list again.  I've verified that everything is set yes for search in the site settings.  Surely I should be able to accomplish this, right? 

I want this list to be searchable and accessible only by the owners group.  It should not show in search for members or visitors, and should not be accessible by those groups either.  Any help is greatly appreciated.