The author studies operator equations of the type \[ T' Tu+ \varphi(\lambda)N(u)=\lambda u, \] where \(T\) is a densely defined closed linear operator in a Hilbert space \(H\), and \(N\) is a continuous nonlinear operator of potential type in \(H\). In particular, spectral gaps in the spectrum of the operator \(T^* T\) are analyzed. The abstract results are illustrated by an application to the nonlinear Schrödinger equation \[ - \Delta u+ q(x)u+\varphi(\lambda)f(x,u)=\lambda u, \] where \(q\) is an \(L^ 2\)-potential and \(f\) is a Carathéodory function satisfying appropriate sign and growth conditions.