Consider the solution of the free time-dependent Schrödinger equation with initial data f. It is shown by Sjögren and Sjölin that there exists f in the Sobolev spaceHs (Rd ), s = d/2 such that tangential convergence can not be widened to convergence regions. The author obtained in a previous paper the corresponding results for a generalized version of the Schrödinger equation, where −Δx is replaced by an operator ϕ(D), with special conditions on ϕ. In this paper we show that similar results may be obtained for initial data in usual and mixed Fourier Lebesgue spaces. We also relax the conditions on ϕ.