In order to make sure that your custom list is visible to the SharePoint search results for the anonymous users, you need to set the AnonmyousPermMask64 property for that particular list.
The whole list of the powershell commands is given as follows.
$web = get-spweb “http://yoursiteurl/subweb/subsubweb”
$mylist = $web.Lists["ListName"]
$mylist.BreakRoleInheritance($true)
$mylist.AnonymousPermMask64= $list.AnonymousPermMask64 -bor ([int][Microsoft.SharePoint.SPBasePermissions]::AnonymousSearchAccessList)
$list.Update()
Happy SharePointing :-)

Lakshmanan Sethu SankaranarayanPosted Mar 30, 2015, 3:02 AM
Thank you Andrew Robinson. I m happy that my blog helped you to meet deadline :-)
Andrew RobinsonPosted Mar 27, 2015, 8:44 PM
thankyou, after 3 hrs of reading and trying posts about queryparametertemplate.xml and feeling like I was bashing my head against a brick wall, this was the fix I needed. Much appreciated! My clients new public facing website will now have image spotlights rolling up on their pages when they go live next week :)