Positive Perturbations of Unbounded Operators

Positive Perturbations of Unbounded Operators

This work studies the spectral properties of certain unbounded selfadjoint operators by considering positive perturbations of such operators and the unitary equivalence of the perturbed and unperturbed transformations. Conditions are obtained on the unitary operators implementing this equivalence which guarantee that the selfadjoint operators have an absolutely continuous part.

Unbounded Perturbations of the Bose-einstein Condensate

New, commutative probability tools for studying the problem of the Bose-Einstein condensate preservation under thermooeld and standard gauge-invariant perturbations are presented. In particular, a new result on the stability of a Bose-Einstein condensate under the thermooeld perturbations of a polynomial type is presented. The problem of the Bose-Einstein condensate preservation that do arises ...

Differentiable Perturbation of Unbounded Operators

If A(t) is a C1,α-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized C1 in t. If A is C∞ then the eigenvalues can be parameterized twice differentiable. Theorem. Let t 7→ A(t) for t ∈ R be a curve of unbounded self-adjoint operators in a Hilbert space with common domain of definition and with compact resol...

Perturbations of Operators, Ii

Unbounded operators, Friedrichs’ extension theorem

Explicit naming of the domain of an unbounded operator is often suppressed, instead writing T1 ⊂ T2 when T2 is an extension of T1, in the sense that the domain of T2 contains that of T1, and the restriction of T2 to the domain of T1 agrees with T1. An operator T ′, D′ is a sub-adjoint to an operator T,D when 〈Tv,w〉 = 〈v, T ′w〉 (for v ∈ D, w ∈ D′) For D dense, for given D′ there is at most one T...