Prescribed spectral configurations under rank-one perturbations
Abstract
A spectral configuration specifies the Fredholm set, the Fredholm index on each of its components, and, on zero-index components, either the whole component or a locally finite set of discrete eigenvalues with prescribed algebraic multiplicities. We show that any two spectral configurations with the same Fredholm set and index can be realized by a suitable closed operator A and a rank-one perturbation A+K. This remains true even when the rank-one operator K, acting on a given separable infinite-dimensional Hilbert space, is prescribed in advance. In particular, for any such prescribed K, one can choose A such that the discrete eigenvalues of A+K accumulate at the common essential spectrum at any prescribed rate. The proof is inspired by classical linearization theory. The main tool is the fact that arbitrary holomorphic functions can be represented as rank-one perturbation determinants.
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BibTeX
@misc{hansmann2026prescribed,
title = {{Prescribed spectral configurations under rank-one perturbations}},
author = {Marcel Hansmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33630},
eprint = {2610.33630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33630},
}
RIS
TY - UNPB
TI - Prescribed spectral configurations under rank-one perturbations
AU - Marcel Hansmann
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33630
UR - https://arxiv.org/abs/2610.33630
AB - A spectral configuration specifies the Fredholm set, the Fredholm index on each of its components, and, on zero-index components, either the whole component or a locally finite set of discrete eigenvalues with prescribed algebraic multiplicities. We show that any two spectral configurations with the same Fredholm set and index can be realized by a suitable closed operator A and a rank-one perturbation A+K. This remains true even when the rank-one operator K, acting on a given separable infinite-dimensional Hilbert space, is prescribed in advance. In particular, for any such prescribed K, one can choose A such that the discrete eigenvalues of A+K accumulate at the common essential spectrum at any prescribed rate. The proof is inspired by classical linearization theory. The main tool is the fact that arbitrary holomorphic functions can be represented as rank-one perturbation determinants.
ER -