On the Right Hamiltonian for Singular Perturbations: General Theory

Let a pair of self-adjoint operators fA 0;W 0g be such that (a) there is a dense domain D dom(A) \ dom(W ) that _ H = (A +W )jD is semibounded from below (stability domain), (b) the symmetric operator _ H is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension  of _ A = AjD is maximal with respect to W , i.e., dom(p W )\ ker( _ A I) = f0g, < 0. Let fWng1n=1 be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W . The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians ~ Hn = ~ A+Wn exists and is unique for any self-adjoint extension ~ A of _ A, (ii) to describe the limit H. We show that under the conditions (a) (c) there is a regularizing sequence fWng1n=1 such that ~ Hn = ~ A +Wn tends in the strong resolvent sense to unique (right Hamiltonian) Ĥ =  : + W , otherwise the limit is not unique.