Singular Perturbations of Self-Adjoint Operators

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A : D(A) ⊆ H → H be an injective self-adjoint operator and let τ : D(A) → X, X a Banach space, be a surjective linear map such that ‖τφ‖X ≤ c ‖Aφ‖H. Supposing that Range (τ ) ∩ H = {0}, we define a family AτΘ of self-adjoint operators which are extensions of the symmetric operator A|{τ=0} . Any φ in the operator domain D(A τ Θ) is characterized by a sort of boundary conditions on its univoc...

On Eigenvalues Problem for Self-adjoint Operators with Singular Perturbations

We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbations. The general results presented here include weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so-called additive strongly singular perturbations.

Rank one perturbations of self-adjoint operators

A Krein-like Formula for Singular Perturbations of Self-adjoint Operators and Applications

Given a self-adjoint operator A : D(A) ⊆ H → H and a continuous linear operator τ : D(A) → X with Range τ ′ ∩ H = {0}, X a Banach space, we explicitly construct a family A Θ of self-adjoint operators such that any A Θ coincides with the original A on the kernel of τ . Such a family is obtained by giving a Krĕın-like formula where the role of the deficiency spaces is played by the dual pair (X ,...

Boundary Triples and Weyl Functions for Singular Perturbations of Self-adjoint Operators

Given the symmetric operator AN obtained by restricting the self-adjoint operator A to N , a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A N and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of AN and their resolvents.