I am a postdoc currently on the academic job market who does interdisciplinary work. My phd is in mathematical physics, granted by a physics department, while working with a research group technically within a computer science department. I do work in quantum computation and lean heavily towards mathematics, unlike many researchers in QC. I publish in all of mathematics, physics, and computer science. There is a body of mathematical knowledge lacking in the topic that I study and, therefore, I work on both building and exploiting the necessary math from the ground up.
I'm reworking my research statement and will be applying to positions in all of mathematics, physics, and computer science. Should I be writing different research statements targeted at each department or will one statement that summarizes my overarching project suffice? Also, will describing myself as a mathematical physicist (as opposed to a mathematician or computer scientist) hurt me when applying to jobs outside of physics proper.