  Although energy levels are often given by solutions of the radial equation
such that u(0) is non zero, and hence by first-order singular functions which
are not eigenfunctions of H, the latter is always considered as the only
operator that gives energy levels. Vibrational levels of diatomic molecules are
a usual example. We show that the operator which has singular eigenfunctions,
or pseudofunctions, that give energy levels, is the operator whose action on
pseudofunctions amounts to the embedding in the distributions of R3 of their
Hamiltonian in R3/{0}. When its eigenfunctions are regular, this operator
amounts to H. Energy levels, which are given by eigenfunctions of H when u(0)
is zero, are thus given in any case by eigenfunctions of this operator, which
is an extension of the Schrodinger operator, but not of the Hamiltonian.
