Consider the solution of the time-dependent Schrödinger equation with initial data $f$. It is shown by Sjögren and Sjölin (1989) that there exists $f$ in the Sobolev space $H^s(\mathbf R^n)$, $s=n/2$ such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when $-\Delta_x$ is replaced by an operator $\varphi(D)$, with special conditions on $\varphi$.