My question is specific to admissions in the graduate mathematics. I have had an unconventional training this far; I started off as an Economics major. I realized that I enjoy doing physics and mathematics much more, especially when I had my first exposure to a proof based course in mathematics. I then started taking physics and mathematics. Unfortunately, though, I took more physics than mathematics courses, even though I seem to be more interested in mathematics.
I am contemplating applying to graduate school in mathematics, hoping to work under the supervision of someone who works in mathematical physics. Most mathematics departments mention that 3 letters of recommendations are required. Here's my situation:
I can arrange one mathematics professor, say Y, to write a letter for me. I have taken a second course in analysis with her, and I have been her TA for the first course in analysis twice. Other than that, due to my unconventional background, I have been studying mathematics with her out of class, and I continue to do so during the next year which I shall be taking off. My other two letter writers will be physicists: one works in the field of quantum dynamics; by the time I graduate, I would have had 4 courses and a paper with him, though I don't see myself working in his area of inquiry. The other one will be, hopefully, from a high energy physicist, with whom I took 2 400 level physics courses, and studied a bit of QFT in the summer. I'm not sure if I will continue working with starting summer as he is due to leave for a sabbatical.
Should I arrange for more letter writers who are mathematicians? If so, what difference would that make in my case? For instance, suppose I arrange letter write X. With X, the best I can do in the next few months is to cover some material that I haven't done in a course. I can do the same material with professor Y as well, who seems open to the idea of my discussing the material I self study with her. Or, should I contact another instructor, X, and perhaps try and do something else with him or her. If so, what options should one choose to strength the application?