Any time you are changing URLs of a site you have to be careful to make sure that both Google and Bing know that it is the same site with a different address. You do that by putting in 301 redirects, rel canonical tags, and adding the new site in Webmaster Tools.
Google Webmaster Tools is pretty straight forward. Google has a nice site write up here.
Bing is a little bit different. Both Bing and Google have a site move tool but with Bing you don’t need to use the site move tool if you are just moving to https. When I contacted Bing I received this-
“Bing Webmaster Tools will identify your site thru the domain therefore the SSL and non-SSL version of the site is basically the same in Bing Webmaster Tool. If you already have 301 redirects set to your pages then there’s no need to use the site move feature. The permanent redirect that you’ve placed should already be sufficient for Bing to change your site to http to https.”
So make sure you have your 301s in and as long as your domain is staying the same you are ok just leaving everything the same in Bing Webmaster Tools.