Hi Artus, Mathematical Economics is not a separate branch of Economics like Macroeconomics, Microeconomics, International Trade, etc. It is just a way of economic analysis, where the mathematical proofs and theorems are used heavily to make the economic models, for instance Ct + It = Yt = F(Kt,L), It = Kt+1 - (1-δ)Kt, It = sF(Kt,L)
is the most fundamental model in macroeconomics called the Solow's Growth Model, which was build upon using mathematics, thus it can be called as mathematical economics, eventually economic analysis.
If you are interested doing research in this area i would higly recommend you to visit this site http://econ.lse.ac.uk/study/grpr.html
that is the best-in-the-world economics graduate program in that specific area which would help you get into an elite PhD program anywhere in the world. But you must know that unlike Physics or other Science fields, in economics you can do research in any area of your fancy if you have a good bachelors degree in economics(with a very good math courses in it) and a graduate program in some cases.
And yes, science students do very well in Economics graduate programs, especially Physicists and Engineers.
Good luck, PM me if you need any help!!