Vol. 35, No. 2, 1970

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Almost smooth perturbations of self-adjoint operators

John Ben Butler, Jr.

Vol. 35 (1970), No. 2, 297–306

Assume H0 ∈𝒞(H) is a self-adjoint operator with spectrum on [0,) and that E0(Δ) ∈ℬ(H) is the spectral measure determined by H0,Δ [0,). Let H1 = H0 + V where V = B A and A,B ∈ℬ(H) are commuling self-adjoint operators. In this paper T. Kato’s concept of smooth perturbations is generalized in the following way: H1 is said to be an almost smooth perturbation of H0, except at 1 = 0, if A,B are smooth with respect to H0E0m) for all intervals Δm = (1∕m,),m 1. It is proved that the time independent wave operators corresponding to H0,H1 exist when the assumption that H1 is smooth with respect to H0 is replaced by the assumption that H1 is almost smooth with respect to H0.

Mathematical Subject Classification
Primary: 47.48
Received: 19 November 1969
Published: 1 November 1970
John Ben Butler, Jr.