The von Neumann problem and singular perturbations of nonnegative operators

Eduard Tsekanovskii Niagara University, USA

We develop a new approach and present a solution to von Neumann’s problem on parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator. Our formulas are based on the Friedrichs extension and also provide a description for closed sesquilinear forms associated with nonnegative self-adjoint extensions. All basic results of the wellknown Kreĭn and Birman-Vishik theory and its complementations are derived as consequences from our new formulas, including the parametrization in terms of classical von Neumann’s formulas of all domains of nonnegative self-adjoint extensions as well as canonical resolvents. As an application all nonnegative quantum Hamiltonians corresponding to point-interactions in ℝ^{3} are described. For a nonnegative self-adjoint operator A we establish for the first time necessary and sufficient conditions on singular perturbation G that results the existence of nonnegative selfadjoint operators among the realizations of A+G, as well as when those nonnegative realizations contain the Krein-von Neumann nonnegative extremal self-adjoint extension. Singular perturbations of the Laplace operator in ℝ^{3} by delta potentials are considered. The talk is based on joint work with Yu.Arlinskii.