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We establish the existence of a nontrivial weak solution to stronglyindefinite asymptotically linear and superlinear Schr\"odinger equations. Thenovelty is to identify the essential relation between the spectrum of theoperator and the behavior of the nonlinear term, in order to weaken thenecessary assumptions to obtain a linking structure to the problem, forinstance to allow zero being in the spectrum or the nonlinearity beingsign-changing. Our main difficulty is to overcome the lack of monotonicity onthe nonlinear term, as well, as the lack of compactness since the domain isunbounded. With this purpose, we require periodicity on $V$.

A Note on Nonlinear Schr\"odinger Equations: Unveiling the Relation Between Spectral Gaps and the Nonlinearity Behavior