Perturbation of self-adjoint operators by Dirac distributions

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Abstract
The existence of a family of self‐adjoint Hamiltonians Hϑ, ϑ ∈ [0, 2π), corresponding to the formal expression H0+νδ (x) is shown for a general class of self‐adjoint operators H0. Expressions for the Green’s function and wavefunction corresponding to Hϑ are obtained in terms of the Green’s function and wavefunction corresponding to H0. Similar results are shown for the perturbation of H0 by a finite sum of Dirac distributions. A prescription is given for obtaining Hϑ as the strong resolvent limit of a family of momentum cutoff Hamiltonians HN. The relationship between the scattering theories corresponding to HN and Hϑ is examined.