Resonances under rank-one perturbations

Rank one perturbations of self-adjoint operators

Rank One Perturbations with Infinitesimal Coupling

We consider a positive self-adjoint operator A and formal rank one pertubrations

Bounded Rank-one Perturbations in Sampling Theory

Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from a family of bounded rank-one perturbations of a self-adjoint operator that has only discrete spectrum of multiplicity one. Mathematics Subject Classificatio...

Rank One Perturbations at Infinite Coupling

We discuss rank one perturbationsAα = A+α(φ, ·)φ,α ∈ R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα − z)−1φ) = ∫ dμα(x)/(x− z). We prove there is a measure dρ∞ which is the weak limit of (1 + α)dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent limit of Aα, is unitarily equivalent to multiplication by x on L(R , dρ∞). This generalizes results kno...

Rank One Perturbations, Approximations, and Selfadjoint Extensions

of a semibounded selfadjoint operator A are studied with the help of distribution theory. It is shown that such perturbations can be defined for finite values of : even if the element . does not belong to H&1(A). Approximations of the rank one perturbations are constructed in the strong operator topology. It is proven that rank one H&2 perturbations can be defined uniquely for the homogeneous o...