Bounds on Perturbations of Self-Adjoint Operators

Rank one perturbations of self-adjoint operators

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.

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...

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...

Local two-sided bounds for eigenvalues of self-adjoint operators

We examine the equivalence between an extension of the Lehmann-Maehly-Goerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We establish a general framework which allows sharpening various previously known results in these two settings and determine explicit convergence estimates for both methods. We...