The functional-analytic versus the functional-integral approach to quantum Hamiltonians: The one-dimensional hydrogen atom.

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

14 pages, RevTeX 3.0, AMSfonts, 1 Figure available upon request from the Authors, to appear in J. Math. Phys. May 1995. Changes: Eq. (3) is rewritten and it is clarified that no weak derivatives are used

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