On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A0 acting on a Hilbert space H singular perturbations of the form A0 + V , V = n1 bij ψj , · ψi are studied under some additional requirements of symmetry imposed on the initial operator A0 and the singular elements ψj . A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A0+V . The results are applied for the investigation of singular perturbations of the Schrödinger operator in L2(R3) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.