The functional–analytic versus the functional–integral approach to quantum Hamiltonians: The one-dimensional hydrogen atoma)

Abstract

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 modeling certain experimental situations as, for instance, in quantum wires.