I am a graduate student in mathematics working on functional analysis of quantum information (basically using matrix techniques in some well-known problems). I am presently applying for postdoctoral positions (in both Mathematics and Quantitative Physics departments). However, I am not getting any reply from most of the places and got a few rejection letters as well. Since I am from Mathematics, I cannot apply to experimental groups (where there are more openings). Again, being a Mathematics student and working in Physics (even though there are obvious connections between them) is not seen as a good example among certain parts of the physics community and the majority of the Mathematicians in this part of the world (personal opinion based on certain bitter experiences). I do not know whether it is the same in other places as well.
I feel that I have not projected myself by writing a good research statement. I have written it after reading the first few entries of Google search results as well as advice from the AMS. I have mostly highlighted my present work, with only a short discussion of future ideas.
To make my life more complicated, I want to move out of the present problems which I am working presently and choose some new problems for my postdoctoral research. I have a peripheral knowledge about the works of different groups. However, modifying my research statement according to the research work of each group is, I think, difficult (as I need to spend time to read their works and spot exactly where my knowledge can be used in their problems and so on). The reason for not having time is I am simultaneously working on some research problem as well as writing my thesis, which I need to submit in the coming months.
In this situation, how do I write a good research statement? I want to convey the message to potential employers that, while I have not done research on their problem yet (though I know them to certain level), given an opportunity I can learn and do work in their areas.