Best practice SEO: move domain from http to https (with completely new links)?


I have an old (ugly, completely based on a special Ajax middle ware, without any mobile support, and so on) app, running on a domain www.xx.ch under http.
I’m on the way to develop a complete new app, that will run under the same domain name (www.xx.ch) but under https (with public cert, data compression, responsive, and so on).
The new app will have completely new (different) links that I then will provide to to crawlers in a (completely different) sitemap.
The old (actual running) app has only a few SEO entries.
The idea is to simply stop the old app (no redirection) and run the new app after the migration (so.. if a user call www.xx.ch, the new app will be started under https (and the old one will not be reachable anymore).

The (only few) SEO entries to the old app don’t care, but it’s important, that the new links are scanned on an optimal way by the spiders. I then will generate a new sitemap and upload it for scanning (at least to google and bing).
Questions:

  • Do the spiders separate SEO entries to http and https (same domain name)?
  • Do I have to delete the old entries to the http links (to prevent negative side effects) and – if yes – what is the best practices to do that?
  • Do I have to do more for a good SEO result – if yes what?

Thanks for any help.