Single user mode effectively means that one and a max of only one connection can be made to the database. Do realize this before trying anything of this sort out. This also means that once a user has connected to the database, YOU cannot connect to the database even for admin tasks.
This is just idle curiousity, but why on earth would you want such a thing ?!?!
When I think about the kind of effort I spend getting more users into the same data at the same time, it is hard for me to imagine why someone might deliberately choose to make a database single user. It isn't hard to do, I just can't imagine what reason might provoke you into doing it!
If you'd like to have the same user which is used to connect to the database specific for this application, you could create a sqluser and use that from the application. You'll have to think about how to handle the password (if any): keep it save and easy to maintain. If it's *one* user for each instance of the application, I guess you'll have to look for connection pooling.