Remarks on solving the one-dimensional time-dependent Schrödinger equation on the interval : the case of a quantum bouncer

It is shown that the 1D Hamiltonian, which is a sum of operators which generate a finite nilpotent Lie algebra and depends explicitly on time existing closed form solutions of the time-dependent Schrodinger equation, cannot fulfil in general boundary and normalization conditions on a positive semi-axis. An explanation of the controversy surrounding the solutions of the quantum bouncer model, which appeared recently in the literature, is given.