Since you talk about 'jump' discontinuities, I guess you are interested in a one dimensional Schroedinger equation, i.e., $x\in\mathbb{R}$. In this situation a nice theory can be developed under the sole assumption that $V\in L^1(\mathbb{R})$ (and real valued of course). By a nice theory I mean that the operator $-d^2/dx^2+V(x)$ is selfadjoint, with continuous spectrum the positive real axis, and (possibly) a sequence of negative eigenvalues accumulating at 0. Better behaviour can be produced by requiring that $(1+|x|)^a V(x)$ be integrable (e.g. for $a=1$ the negative eigenvalues are at most finite in number). If you are interested in this point of view, a nice starting point might be the classical paper by Deift and Trubowitz on Communications Pure Appl. Math. 1979. Notice that the solutions are at least $H^1$ H^1_{loc}$(hence continuous) and even something more. A theory for the case$V$= Dirac delta (or combination of a finite number of deltas) was developed by Albeverio et al.; the definition of the Schroedinger operator must be tweaked a little to make sense of it. This is probably beyond your interests. Summing up, no differentiability at all is required on the potential to solve the equation in a meaningful way. However, I suspect that this point of view is too mathematical and you are actually more interested in the physical relevance of the assumptions. 1 Since you talk about 'jump' discontinuities, I guess you are interested in a one dimensional Schroedinger equation, i.e.,$x\in\mathbb{R}$. In this situation a nice theory can be developed under the sole assumption that$V\in L^1(\mathbb{R})$(and real valued of course). By a nice theory I mean that the operator$-d^2/dx^2+V(x)$is selfadjoint, with continuous spectrum the positive real axis, and (possibly) a sequence of negative eigenvalues accumulating at 0. Better behaviour can be produced by requiring that$(1+|x|)^a V(x)$be integrable (e.g. for$a=1$the negative eigenvalues are at most finite in number). If you are interested in this point of view, a nice starting point might be the classical paper by Deift and Trubowitz on Communications Pure Appl. Math. 1979. Notice that the solutions are at least$H^1$(hence continuous) and even something more. A theory for the case$V\$ = Dirac delta (or combination of a finite number of deltas) was developed by Albeverio et al.; the definition of the Schroedinger operator must be tweaked a little to make sense of it. This is probably beyond your interests.