Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

Abstract

Abstract. We analyze the perturbations T +B of a selfadjoint operator T in a Hilbert space H with discrete spectrum {t k } , T φ k = t k φ k , as an extension of our constructions in [1] where T was a harmonic oscillator operator. In particular, if t k+1 − t k ≥ ck α−1 , α > 1/2 and Bφ k = o(k α−1 ) then the system of root vectors of T + B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H. Statement of main resultsLet H be a separable Hilbert space. Consider an operator T with domain domT whose spectrum consists of a countable set of eigenvalues τ = {t k } ∞ k=1 with corresponding eigenvectors {φ k },which form an orthonormal basis in H. Let us also assume that t k+1 −t k > 0 and that for some fixed p ∈ Z + , d > 0(1.1) Define △t k = t k+1 −t k . Then (1

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

“…It would be natural to expect that Hamiltonians whose imaginary parts are integrable functions on phase space will not lead to similar divergences, and, in the presence of P T -symmetry, a bounded mapping to a Hermitian counterpart would be expected. It has been proven in [31][32][33], for example, that certain well-behaved types of non-Hermitian perturbations of Hermitian operators with "harmonic oscillator type" spectra will lead to Riesz bases, an no finite time divergences would be expected. The systematic investigation of the connection between such mathematical constraints and their physical interpretation would be an interesting topic for further studies.…”

On the completeness of root vectors generated by systems of coupled hyperbolic equations

“…In H we consider a matrix differential operator L αβ , where α and β are the two-component vectors, α = α (1) , α (2) , β = β (1) , β (2) . This operator is defined by matrix differential expression (4.8) on the following domain: In the sequel, it is convenient to represent the operator L αβ as the sum, L αβ = L αβ + M, where L αβ coincides with L αβ if one sets C 1 = C 2 = 0.…”

“…Our main result of [41] is the following statement: Theorem 4.4 Assume that the structural parameters are such thatk 1 =k 2 =k andk 1 =k 2 =k. Assume that the boundary parameters α (i) and β (i) , i = 1, 2, are such that the following conditions hold: α (1) = α (2) , β (1) = β (2) ,…”

“…The second paper generalizes the results of the first one to the case of a specific perturbation of a selfadjoint operator in a Hilbert space. The authors of prove the basis property results by employing the spectral asymptotics of the unperturbed selfadjoint operator and evaluating the action of the perturbation operator on the eigenfunctions of the original selfadjoint operator. Among other results, the authors give an elegant construction of a non‐selfadjoint operator with a discrete spectrum, whose set of the generalized eigenfunctions is minimal and complete in a Hilbert space, but is not uniformly minimal.…”

Local form-subordination condition and Riesz basisness of root systems

“…Regarding the relations to previous works, the special version of the condition (1.10) with γ = 1 was introduced in [28]; the relation to the operator version of (1.10) used in [2,3,33,34] is discussed in [28] as well. Comparing to previous papers, we allow here a faster condensation of {µ k } at infinity, namely µ k ∼ k γ with γ > 0 is possible, cf.…”