Bound-state QED calculations and the hydrogen molecular ion spectroscopy

In the last few years we achieved substantial progress in determination of transition frequencies in the hydrogen molecular ions H+2 and HD+. Particularly, the fundamental transitions for these ions were calculated with the fractional uncertainty of 7:5 10?12.In our presentation we want to discuss the following topics: Numerical calculations of contributions of orders m7 and m8.The contributions of orders m6 and higher have been obtained in the nonrecoil limit. Still our recent work [1] shows that some care should be taken in order to properly evaluate a complete contribution. We suggest a new formalism basedon the adiabatic multichannel approximation, which allows to pose the problem in a more rigorous way. This new approach makes possible both to calculate the nonrecoil corrections of orders m6 and higher and to get estimates on the
nonadiabatic contributions, or in other words to set properly error bars of our calculation.As an example of application of this new formalism we want to consider relativis-tic corrections of order m6. New values for the hyperne structure intervals will be presented, which allow to bring agreement with precision experimental data by Jeerts [2] to a level of 1 ppm.Finally, we would like to discuss various applications of precision spectroscopy of light atoms and molecules. Particularly, its contribution to the fundamental physical constants, molecular optical clocks, tests of CPT invariance, etc.