Non-symmetric perturbations of self-adjoint operators

Non-negative Perturbations of Non-negative Self-adjoint Operators

Let A be a non-negative self-adjoint operator in a Hilbert space H and A0 be some densely defined closed restriction of A0, A0 ⊆ A 6= A0. It is of interest to know whether A is the unique non-negative self-adjoint extensions of A0 in H. We give a natural criterion that this is the case and if it fails, we describe all non-negative extensions of A0. The obtained results are applied to investigat...

Rank one perturbations of self-adjoint operators

Non-self-adjoint Differential Operators

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

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.