Between 1971 and 1973, Böttcher was in an elite class for mathematics at the Chemnitz University of Technology. In 1973, he won a silver medal at the International Math Olympiad in Moscow. He studied mathematics from 1975 to 1979 in Chemnitz. In 1980, he went to Rostov on Don to study and earned a PhD there in 1984. The topic of his dissertation was The finite section method for the Wiener-Hopf integral operator. After this he worked as scientist assistance at the university of Chemnitz. Since 1992, he is professor for harmonic analysis and Operator theory at the Chemnitz University of Technology.
As of 2012, he is the author of nine books and about 180 papers.