We recently began using a load balancer to force HTTPS on our SharePoint, and the PowerShell code below can no longer find the site. I've tried using both https://mysite and //mysite, with the same results.
[void][System.reflection.Assembly]::LoadWithPartialName("Microsoft.SharePoint")
$site = new-Object Microsoft.SharePoint.SPSite("http://mysite")
$web = $site.rootweb
$list = $web.Lists["mylist"]