  The capabilities of the functional-analytic and of the functional-integral
approach for the construction of the Hamiltonian as a self-adjoint operator on
Hilbert space are compared in the context of non-relativistic quantum
mechanics. Differences are worked out by taking the one-dimensional hydrogen
atom as an example, that is, a point mass on the Euclidean line subjected to
the inverse-distance potential. This particular choice is made with the intent
to clarify a long-lasting discussion about its spectral properties. In fact,
for the four-parameter family of possible Hamiltonians the corresponding
energy-dependent Green functions are derived in closed form. The multiplicity
of Hamiltonians should be kept in mind when modelling certain experimental
situations as, for instance, in quantum wires.
