  We derive a mean-field relativistic formula for the isotope shift of an
electronic energy level for arbitrary angular momentum; we then use it to
predict the spectra of superheavy metastable neutron-rich isotopes belonging to
the hypothetical island of stability. Our results may be applied to the search
for superheavy atoms in astrophysical spectra using the known values of the
transition frequencies for the neutron deficient isotopes produced in the
laboratory. An example of a relevant astrophysical system may be the spectra of
the Przybylski's star where superheavy elements up to Z=99 have been possibly
identified. In addition, it has been recently suggested to use the measurements
of King plot non-linearity in a search for hypothetical new light bosons. On
the other hand, one can find the non-linear corrections to the King-plot
arising already in the Standard Model framework. We investigate contributions
to the non-linearity arising from relativistic effects in the isotope
field-shift, the nuclear polarizability and many-body effects. It is found that
the nuclear polarizability contribution can lead to the significant deviation
of the King plot from linearity. Therefore, the measurements of the
non-linearity of King plots may be applied to obtain the nuclear polarizability
change between individual isotopes. We then proceed with providing a rough
analytical estimate of the non-linearity arising solely from the effect of a
hypothetical scalar boson. Our predictions give theoretical limitations on the
sensitivity of the search for new interactions and should help to identify the
most suitable atoms for corresponding experiments.
