ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS - GENERAL-THEORY

Let the pair of self-adjoint operators {A greater than or equal to 0,W less than or equal to 0} be such that: (a) there is a dense domain D subset of or equal to dom(A) boolean AND dom(W) H-. = (A + W)\ D is se mibounded from below (stability domain), (b) the symmetric operator H- . is not essentially self-adjoint (singularity of the perturbation), ( c) the Friedrichs extension (A) over cap of A(.) = A \ D is maximal wi th respect to W, i.e., dom(root-W) boolean AND ker (A - eta I) = {0}. eta < 0. Let {W-n}(infinity)(n=1) be a regularizing sequence of bound ed operators which tends in the strong resolvent sense to W. The abstr act problem of the right Hamiltonian is: (i) to give conditions such t hat the limit H of self-adjoint regularized Hamiltonians (H) over tild e(n) = (A) over tilde + W-n exists and is unique for any self-adjoint extension (A) over cap of A, (ii) to describe the limit H. We show tha t under the conditions (a)-(c) there is a regularizing sequence {W-n}( infinity)(n=1) such that H-n = (A) over tilde + W-n tends in the stron g resolvent sense to unique (right Hamiltonian) (H) over cap = (A) ove r cap + W, otherwise the limit is not unique.