The abstract Birman-Schwinger principle and spectral stability

Abstract

We discuss abstract Birman-Schwinger principles to study spectra of self-adjoint operators subject to small non-self-adjoint perturbations in a factorised form. In particular, we extend and in part improve a classical result by Kato which ensures spectral stability. As an application, we revisit known results for Schrödinger and Dirac operators in Euclidean spaces and establish new results for Schrödinger operators in three-dimensional hyperbolic space.

Keller-type bounds for Dirac operators perturbed by rigid potentials

“…The above result correspond to Theorem 5 in [32], where the operator H V is obtained via the pseudo-Friedrichs extension. Instead, from Theorem 6, 7, 8 and Corollary 4 of [32] we get the abstract Birman-Schwinger principle stated below. Lemma 5.…”

“…The number of works in the literature which make use of the Birman-Schwinger principle is huge. Anyway, recently an abstract analysis has been carried out by Hansmann and Krejčiřík in [32]. Here we sketch in a synthetic way their approach.…”

“…By strengthening Assumption II asking the norm of the Birman-Schwinger operator to be strictly less than 1 uniformly respect to z ∈ ρ(H 0 ), Theorem 3 in [32] state the following result on the spectrum invariance for the perturbed operator. Lemma 6.…”

Localization of eigenvalues for non-self-adjoint Dirac and Klein-Gordon operators

“…In this section, we recall the technicalities of the Birman-Schwinger principle and we properly define an operator perturbed by a factorizable potential. We rely completely on the abstract analysis carried out by Hansmann and Krejčiřík in [24], to which we refer for more results and background. There, in addition to the point spectrum, appropriate versions of the principle are stated even for the residual, essential and continuous spectra.…”

“…In particular, the eigenvalues of the perturbed operator H are confined in the complex region defined by 1 ≤ K z and the point spectrum is empty if K z < 1 uniformly with respect to z. We refer to [24] for an abstract spectral-theoretic machinery based on the Birman-Schwinger idea.…”