Lindau was awarded his PhD in 1971 at Chalmers University of Technology with his dissertation about photoemission and optical absorption studies of the band structure. After the dissertation, he began working at the Silicon Valley-based company Varian Associates in Palo Alto between 1971 and 1972 before he was employed at Stanford University in 1972. In 1973 he obtained the first X-ray photoemission spectra of the 4 f levels of gold. In 1980 he became a professor at Stanford University for his research in electrical engineering and photonics. During his professorships he took a sabbatical at Lund University between 1988 and 1989, where he worked at MAX Lab to expand its research facility. After his sabbatical ended he return to Lund University to become a professor for his research in synchrotron light physics. In 1991, Lindau succeeded Bengt Forkman as director of the Max Lab. As a director, he was in charge of the creation of the second accelerator for synchrotron radiation research in MAX Lab.