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The stability bounds and error estimates for a compact higher orderNumerov-Crank-Nicolson scheme on non-uniform space meshes for the 1Dtime-dependent Schr\"odinger equation have been recently derived. This analysishas been done in $L^2$ and $H^1$ mesh norms and used the non-standard"converse" condition $h_\omega\leq c_0\tau$, where $h_\omega$ is the mean spacestep, $\tau$ is the time step and $c_0>0$. Now we prove that such condition isnecessary for some families of non-uniform meshes and any space norm. Alsonumerical results show unacceptably wrong behavior of numerical solutions(their dramatic mass non-conservation) when this condition is violated.

A "converse" stability condition is necessary for a compact higher order scheme on non-uniform meshes