Singularly Perturbed Operators of Type -Δ + λδ

Abstract

Let A = A * be a self-adjoint unbounded positive operator in a separable complex Hilbert space ℋ and let ℋ_⊃ ℋ⊃ ℋ+ be the rigged Hilbert space associated with A, i.e. ℋ+ coincides with the domain ℱ(A) in the graph-norm of A and ℋ_ is the conjugate space to ℋ + . By analogy with the Schrodinger operator -Δλ,δ =-Δ+λδ with the singular δ-potential we consider the formal expression A = A λ,ω=A+λω, λ∈ℝ1,ω∈ℋ_\ℋ and give for it sense of a self-adjoint operator in ℋ. Our approach is based on the self-adjoint extensions theory of symmetric operators. We call the constructed operator A λ,ω, by singularly perturbed with respect to A and study its spectral properties.