Landau relationship for the two-side profile of the one-dimensional potential distribution of the mechanical oscillator with the predefined dependence of the period of oscillations from energy

The Landau relationship is considered which enables to reconstruct the symmetrical potential well for the one-dimensional mechanical oscillator using the predefined dependence of its period from energy as the input data. It is shown how this expression can be modified to produce non-symmetrical smooth potential wells for the same purpose. It is confirmed that there are no one-dimensional potential distributions with flat bottoms and/or with non-monotonic behavior which produce the ideally isochronous oscillators (i.e., the oscillators where the period of oscillations is the same for any energy at least for some energy ranges). The application of these results to time-of-flight mass spectrometers and FTR-MS electrostatic traps is discussed.