Direct and Inverse Spectral Problems for Rank-One Perturbations of Self-adjoint Operators

Abstract

For a given self-adjoint operator A with discrete spectrum, we completely characterise possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of reconstructing B from its spectrum.

Reconstruction of Differential Operators with Frozen Argument

“…By studying the latter, we completely characterize eigenvalue asymptotics as stated in Theorems 1 and 2. We stress that this asymptotics differs from the one derived in [13] for the bounded case ϕ, ψ ∈ H, and its derivation requires essential changes in the proofs. Next, zeros of F allow for a unique reconstruction of F, thus specifying ψ up to an iso-spectral set, see Corollary 5.…”

“…In this section, we collect some properties of rank-one perturbations of self-adjoint operators A acting in a fixed separable (infinite-dimensional) Hilbert space H established in [12,13] that will be used to prove the main results of this work. The reader can find further references and examples of applications in the monographs [10,11].…”

“…We should point out the effect on the asymptotic distribution of eigenvalues that singularity of ϕ makes: for regular ϕ ∈ H, the sequence of µ n − λ n was absolutely summable ( [13], Theorem 3.1), while here it is only square summable.…”

“…Generic bounded rank one perturbations of self-adjoint operators are studied quite well, see e.g., the reference lists of [10][11][12][13]. In our resent work [12,13], we gave a complete characterization of possible spectra σ(B) of bounded non-self-adjoint rank-one perturbations…”

“…The paper is organized as follows. In the next section, we introduce necessary definitions and specify the setting of the paper, recall auxiliary results, derive the characteristic function and explain why the results of [12,13] are applicable here. In Section 3, we establish the asymptotics of the eigenvalues of B and in Section 4, we solve the inverse problem of reconstructing the vector ψ given ϕ.…”