  We establish the existence of a nontrivial weak solution to strongly
indefinite asymptotically linear and superlinear Schr\"odinger equations. The
novelty is to identify the essential relation between the spectrum of the
operator and the behavior of the nonlinear term, in order to weaken the
necessary assumptions to obtain a linking structure to the problem, for
instance to allow zero being in the spectrum or the nonlinearity being
sign-changing. Our main difficulty is to overcome the lack of monotonicity on
the nonlinear term, as well, as the lack of compactness since the domain is
unbounded. With this purpose, we require periodicity on $V$.
