Singular perturbations in the interaction representation

A general method for treating highly singular perturbations V of self-adjoint operators H in Hilbert space is applied to the case of perturbations of $\text{(?}\text{id}\text{dx}\text{)}{}^{\mathrm{n}}$ in L₂ ($\text{R}$¹ by (multiplications by) distributions. A self-adjoint operator H_V that agrees with H + V in the usual sense when V is sufficiently regular, and is moreover a continuous function of V, within the class of distributions under consideration, in the strong operator topology for unbounded self-adjoint operators, is shown to exist. This operator H_V need not be semi-bounded, or determined by a sesquilinear form associated with H + V. The method proceeds by construction of the corresponding unitary propagator in the interaction representation, essentially e^?itHVe^itH, which is shown to be expressible as a uniformly convergent perturbative series for small times.