Spectra of rank-one perturbations of self-adjoint operators

Abstract

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic multiplicity

Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators

“…and thus can be analytically extended to σ 0 (A); we keep the notation F for this extension. As proved in [15], the geometric multiplicity of every eigenvalue µ of B is at most 2; multiplicity 2 is only possible when µ ∈ σ 0 (A), i.e., µ = λ n for some n ∈ I 0 and, in addition,…”

“…and denote by N F the set of zeros of F . This function appears in the Krein resolvent formula for B [9,15], and its zeros characterise the spectrum of B.…”

“…Under such an assumption, every eigenvalue µ of B is geometrically simple. One of the main results of [15] claims that the algebraic multiplicity m of an eigenvalue µ of B coincides with the multiplicity l of z = µ as a zero of F if µ ∈ σ 0 (A) and is equal to l + 1 otherwise. In other words, the common spectrum σ 0 (A) = σ 0 (B) of A and B and the zeros of F completely characterize the spectrum of B, counting with multiplicities.…”

“…Proof of Theorem 4.1. Given any sequence ν of complex numbers satisfying the assumption of the theorem, we construct the meromorphic function F via (15). Next, calculate the residua −c n of F at the points λ n ∈ Λ 1 via ( 17) and define the sequences Since the sequence (c n ) n∈I 1 is summable by Lemma 4.4, it follows that the sequences (a n ) n∈I and (b n ) n∈I belong to ℓ 2 (I).…”

“…[2,9,37]), the question is quite non-trivial even for self-adjoint perturbations of a self-adjoint operator A, and thus necessarily much more complicated for generic rank-one perturbations. In our previous work [15], we described local spectral properties of rank-one perturbations of a self-adjoint operator with discrete spectrum. Namely, it was shown therein that, as in the finite-dimensional case [23], such a perturbation can possess eigenvalues of arbitrarily prescribed multiplicities at any finite set of complex numbers; cf.…”