ABSTRACT
We explore further the conjecture that a precision comparison of the absorption spectra of hydrogen and anti-hydrogen might provide a measurement of (or upper bound to) hypothetical deviations from linearity in quantum mechanics. We consider stationary solutions to the family of nonlinear Schrödinger equations proposed earlier by the authors as candidates for a fundamental modification of linear quantum mechanics, with a hydrogenic potential term. Taking into account nonlinear gauge transformations, we obtain (perhaps surprisingly) a gaugeinvariant expression related to the energy levels of bound states as well as plane waves, that is independent of all the proposed sources of nonlinearity. Thus it is possible that this type of nonlinearity would not manifest itself at all as a shift in spectral lines, and could not be observed through the suggested comparison. The general result is independent of the shape of the potential.
