The example they try it on is a quantum harmonic oscillator, i.e. the potential V(x) is just the squared norm of x. That's analytically solvable. Also,

in the case of the Schro ̈dinger equation, due to the separable nature of its network structure, we employ a separate neural network for each dimension.

which doesn't sound very general. Admittedly in my quick skim of this paper I didn't follow much.