I am an international, senior Economics and Mathematics major at middle-sized, private international university. I'm trying to finalize the list of schools to which I will apply (to Math PhD programs in Mathematical Physics), but I am finding it very difficult to contact professors given my unconventional background.
I know that I am not a candidate for admission at top-tier universities. For one, my major isn't either mathematics or physics, even though I have been taking physics and math courses for the last 2 years. Also, I'm sure the professors who will write my recommendation letters aren't very well-known either. I may get to work with a professor who has a good reputation in the US, though. Though this depends if the professor accepts me under his tutelage, since he left my current college some time ago. By the time I graduate, I would have taken a dozen physics courses; I'll be lighter on the mathematics side, though.
I'm not sure if I qualify to be considered as a candidate in the mathematics departments; I want my training to be like a mathematician, but I'd like to work on mathematical physics problems. So, I'm not sure how to gauge how fit I am for a department? To that end, I'd ideally like to email some professors, explaining my unconventional background, express interest in working under them, and get advice if I will be considered in their program.
What's the best way to structure such an email? Should I give an extended summary of my "situation"? Is it common to elicit "advice" from professors; we all know they don't have much time on their hands; do I need to read their papers to contact them before hand? Etc.