Canonical systems and finite rank perturbations of spectra

1996/06/24 by Poltoratski, Alexei G.
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/9606214

Abstract

We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or unitary operators. In the n-dimensional case, we prove that for any cyclic self-adjoint operator A, operator A_λ= A + Σ_k=1ⁿ λ_k(⋅,ϕ_k)ϕ_k is pure point for a. e. λ=(λ₁,λ₂,...,λ_n) ∈\Bbb Rⁿ iff operator A_η=A+η(⋅,ϕ_k)ϕ_k is pure point for a.e. η∈\Bbb R for k=1,2,...,n. We also show that if A_λ is pure point for a.e. λ∈ \Bbb Rⁿ then A_λ is pure point for a.e. λ∈ γ for any analytic curve γ∈\Bbb Rⁿ.

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