Harrell does research on the Schrödinger equation, which describes nonrelativistic matter on the atomic scale, and on similar differential equations. His research has focused primarily on semiclassical quantum mechanics and on the effects of geometry on the eigenvalues of differential equations. He has advised eight PhD dissertations and two MS theses, and has written numerous articles with at least thirty-five coauthors.
Some of Harrell's other passions are travel, world cultures, and science-arts partnerships. He has for example participated in numerous scientific events in developing countries, and has helped create science-themed performances at the Atlanta Science Festival and elsewhere with the production company he co-founded for that purpose, Mathematics in Motion. Harrell has two grown children and lives in Atlanta with his wife, a law professor, when not off in some far-flung part of the world, whether for research and education or to do some trekking in the wildernes.