Consider the solution of the time-dependent Schrodinger equation with initial data f. It is shown by Sjogren and Sjolin (1989) that there exists f in the Sobolev space H-s (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 phi(D), with special conditions on phi.