Searcharxiv⌕ Search

arXiv subjects

Marcel Hansmann

Publications and source records attributed to Marcel Hansmann.

17 recordsLinked to original sources
arXiv2020

The abstract Birman-Schwinger principle and spectral stability

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 Schr\"odinger and Dirac operators in Euclidean spaces and establish new results for Schr\"odinger operators in three-dimensional hyperbolic space.

math.SP↗
arXiv2017

Eigenvalues of compactly perturbed operators via entropy numbers

We derive new estimates for the number of discrete eigenvalues of compactly perturbed operators on Banach spaces, assuming that the perturbing operator is an element of a weak entropy number ideal. Our results improve upon earlier results by the author and by Demuth et al. The main tool in our proofs is an inequality of Carl. In particular, in contrast to all previous results we do not rely on tools from complex analysis.

math.SP↗
arXiv2015

Perturbation determinants in Banach spaces - with an application to eigenvalue estimates for perturbed operators

In the first part of this paper we provide a self-contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of $C_0$-semigroups.

math.SP↗
arXiv2014

Estimating the number of eigenvalues of linear operators on Banach spaces

Let $L_0$ be a bounded operator on a Banach space, and consider a perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with obtaining bounds on the number of eigenvalues of $L$ in subsets of the complement of the essential spectrum of $L_0$, in terms of the approximation numbers of the perturbing operator $K$. Our results can be considered as wide generalizations of classical results on the distribution of eigenvalues of compact operators, which correspond to the case $L_0=0$. They also extend previous results on operators in Hilbert space. Our method employs complex analysis and a new finite-dimensional reduction, allowing us to avoid using the existing theory of determinants in Banach spaces, which would require strong restrictions on $K$. Several open questions regarding the sharpness of our results are raised, and an example is constructed showing that there are some essential differences in the possible distribution of eigenvalues of operators in general Banach spaces, compared to the Hilbert space case.

math.SP↗
arXiv2014

An observation concerning boundary points of the numerical range

A theorem of Hübner states that non-round boundary points of the numerical range of a linear operator, i.e. points where the boundary has infinite curvature, are contained in the spectrum of the operator. In this note, answering a question of Salinas and Velasco, we will show that Hübner's result remains true under the weaker assumption that the boundary has infinite upper curvature. Our short and simple proof relies on some classical ideas of Berberian.

math.SP↗
arXiv2014

On non-round points of the boundary of the numerical range and an application to non-selfadjoint Schrödinger operators

We show that non-round boundary points of the numerical range of an unbounded operator (i.e. points where the boundary has infinite curvature) are contained in the spectrum of the operator. Moreover, we show that non-round boundary points, which are not corner points, lie in the essential spectrum. This generalizes results of Hübner, Farid, Spitkovsky and Salinas and Velasco for the case of bounded operators. We apply our results to non-selfadjoint Schrödinger operators, showing that in this case the boundary of the numerical range can be non-round only at points where it hits the essential spectrum.

math.SP↗
arXiv2013

Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators

Let B=A+K where A is a bounded selfadjoint operator and K is an element of the von Neumann-Schatten ideal S_p with p>1. Let {λ_n} denote an enumeration of the discrete spectrum of B. We show that $\sum_n \dist(λ_n, σ(A))^p$ is bounded from above by a constant multiple of |K|_p^p. We also derive a unitary analog of this estimate and apply it to obtain new estimates on zero-sets of Cauchy transforms.

math.SP↗
arXiv2012

Eigenvalues of non-selfadjoint operators: A comparison of two approaches

The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involving the numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schrödinger operators are considered. Some possible directions for future research are discussed.

math.SP↗
arXiv2009

On the discrete spectrum of non-selfadjoint operators

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schroedinger operators with complex potentials.

math.SP↗
arXiv2009

Inequalities for the eigenvalues of non-selfadjoint Jacobi operators

We prove Lieb-Thirring-type bounds on eigenvalues of non-selfadjoint Jacobi operators, which are nearly as strong as those proven previously for the case of selfadjoint operators by Hundertmark and Simon. We use a method based on determinants of operators and on complex function theory, extending and sharpening earlier work of Borichev, Golinskii and Kupin.

math.SP↗