Exact Results in Quantum Theory

The molecular hydrogen ions (i.e. H2+ and all its isotopologues involving deuterium or tritium) are the simplest molecules, composed of two nuclei and a single electron, which allows their spectrum to be calculated with high accuracy. On the experimental side, ultra-narrow ro-vibrational transitions have been measured at high resolutions, and new techniques under development should allow for further improvement.After presenting the main motivations for precision studies of molecular hydrogen ions, a few recent theoretical results will be discussed. Ongoing work on the one-loop self-energy correction, which is the largest QED contributions to the energy levels and one of the leading sources of theoretical uncertainty, will be presented. I will also describe the calculation of the bound-electron g-factor, which has been recently measured in the rovibrational ground state of HD+.