What to do when a prospective PhD supervisor doesn't follow up after an initial positive response? A few days back I emailed a potential supervisor in one of the Top schools in Computer Science. To my surprise I got an instant reply from him mentioning that I should apply and he has funding for research assistants like me. He also mentioned a few questions about research topics so to follow up I sent him the answer and asked his opinion. However I didn't hear from him after that. Here I would like to mention that questions were not to test me but just general research questions which he is interested in, and my reply was also more like a discussion and asking his opinion. Now I am confused what I can get from this, and also should I ask him somehow to have Skype interview? He is really a great professor and has his own funding so may be he can even effect committee decision.

Thanks.
The professor encouraged you to apply --- so that's what he expects you to do next. It is nothing wrong in your attempt to establish a more substantial connection with him and to promote yourself in his eyes, it increases your chances to get selected. However, imagine that at the same time there are probably 100+ other candidates, trying to do exactly the same --- sending him several smart research questions, involving him in some discussion, and expecting to chat over Skype with him in a few days. How would you feel in his shoes? 

Of course, professors want to be available for their potential PhD students, TAs and colleagues, and of course, they want to participate in the selection process. But many of them do not want to carry out all the communication themselves --- that's what secretariat, admissions, and HRs are for! I assume there is a formal route to submit your application, and I suggest exactly what your professor suggested --- prepare your application and submit as instructed. 

If in a meantime you receive a message from him --- go forth and suggest a Skype meeting. If not --- assume that the professor is busy, and submit your application before the deadline anyway.