  In this paper we investigate when linearly independent eigenfunctions of the
Schr\''odinger operator may have the same modulus. General properties are
established and the one-dimensional case is treated in full generality. The
study is motivated by its application to the bilinear control of the
Schr{\"o}dinger equation. By assuming that the potentials of interaction
satisfy a saturation property and by adapting a strategy recently proposed by
Duca and Nersesyan, we discuss when the system can be steered arbitrarily fast
between energy levels. Extensions of the previous results to quantum graphs are
finally presented.
