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 H0E0(Δm) 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.
