After received my PhD, can I publish a sole-author paper for some extended results? After receiving my Ph.D., I then became self-employed and as a job-seeker. So I use my spare time to continue doing research in the field of my dissertation on my own and came up with some results that I view as an extension of the results which are not included in my thesis nor published anywhere.  Here is my question


  Is it ethical to submit these results as a sole-author paper and perhaps list my advisor name in acknowledgment section for thanking him to bring me into the field ? or should I contact my advisor and asking his permission for publication? 


p.s. I do theoretical research, so there are no experiment issues or budget concerns.
You don't need to ask permission for your ongoing work. It is entirely appropriate to extend what you have done and publish it. You should be sure to cite the older work as needed to avoid questions of self-plagiarism, of course. 

An acknowledgement of your advisor for their past help is a good and appropriate thing to do, but not necessarily required. I would do it unless there were some specific reason not to do so. 

Asking them if they want to participate can also be good, but not at all necessary. It is good simply because having collaborative relationships is good. If you decide to do that, then co-authorship would be a consideration, of course.  But sole-authorship is also proper and is eventually the place (nearly) every researcher will wind up.