This is related to the general question of how to write a proposal to do work that you haven't yet figured out how to do. If you actually tell people what you are thinking of working on you are going sound naive, ignorant or shallow because you don't know how it is going to work out. You could talk about the work you have in preparation (almost done but not quite ready to post or submit yet). But chances are if you are on the job market you are rushing to get your best recent ideas out where people can see them (so they are already posted).

What you are supposed to do in this kind of statement of research interests is first summarize the importance of your work (that means telling people why your thesis is non-trivial and important in a larger scheme). Because this is based on stuff you have done you are going to sound knowledgeable, but more importantly you let people know why your work is not only technically good but important. THEN you briefly mention classes of problems that are related, possible applications or large scale important problems that you might be able to make progress on that are similar or might be solved using the types tools you have developed in your thesis. This part is brief because you don't know yet what you are going to do.

Sometimes people recommend that you look up the fields that people are working on at the institution you are applying to and suggest you might make progress in those or related fields or bring up connections. That takes time but if you know somebody at the institution (and your application isn't in the blind paper bomb category so you have a chance of getting the job) then it might be work the effort. Otherwise you might be recycling your research statement for many applications. In this case a general statement of types of important problems that you might be able to make progress in is the way to go.