arXiv subjects

Sabine Bögli

Publications and source records attributed to Sabine Bögli.

15 recordsLinked to original sources

arXiv2026

On the discrete spectrum of non-selfadjoint operators with applications to Schrödinger operators with complex potentials

For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial trace of the real part of the Birman--Schwinger operator, or an appropriate rotation thereof. While eigenvalue counting estimates of this type are classical in the selfadjoint setting, no analogous connection between the number of discrete eigenvalues and the Birman--Schwinger operator has previously been established in the non-selfadjoint theory. The proof proceeds via techniques in antisymmetric tensor product spaces that serve as a non-selfadjoint replacement for the classical arguments. As an application to Schrödinger operators, we generalise the Cwikel--Lieb--Rozenblum inequality to complex potentials and derive new Lieb--Thirring type inequalities. We also analyse the sharpness of the obtained bounds and discuss their optimality within the considered framework.

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arXiv2025

Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials

We prove optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials. Our results bound eigenvalue power sums (Riesz means) by the $L^p$ norm of the potential, where in contrast to the self-adjoint case, each term needs to be weighted by a function of the ratio of the distance of the eigenvalue to the essential spectrum and the distance to the endpoint(s) thereof. Our Lieb-Thirring type bounds only hold for integrable weight functions. To prove optimality, we establish divergence estimates for non-integrable weight functions. The divergence rates exhibit a logarithmic or even polynomial gain compared to semiclassical methods (Weyl asymptotics) for real potentials.

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arXiv2025

On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials

We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schrödinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.

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arXiv2021

On the asymptotic behavior of solutions to a class of grand canonical master equations

In this article we investigate the long time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitable weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.

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arXiv2021

Improved Lieb-Thirring type inequalities for non-selfadjoint Schrödinger operators

We improve the Lieb-Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schrödinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the one-dimensional case the result is sharp in the sense that if we take a non-integrable function, then an analogous inequality cannot hold.

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arXiv2021

Counterexample to the Laptev--Safronov conjecture

We prove that the Laptev--Safronov conjecture (Comm. Math. Phys., 2009) is false in the range that is not covered by Frank's positive result (Bull. Lond. Math. Soc., 2011). The simple counterexample is adaptable to a large class of Schrödinger type operators, for which we also prove new sharp upper bounds.

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arXiv2020

On Lieb--Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schr{\" o}dinger operators

We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrödinger operators with rectangular barrier potential and complex coupling. Applying the ideas in the continuous setting, we also solve a similar open problem for one-dimensional Schrödinger operators with complex-valued potentials published by Demuth, Hansmann, and Katriel in [Integral Equations Operator Theory 75, No. 1 (2013), 1-5].

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arXiv2019

On the eigenvalues of the Robin Laplacian with a complex parameter

We study the spectrum of the Robin Laplacian with a complex Robin parameter $α$ on a bounded Lipschitz domain $Ω$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on $α\in \mathbb C$, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of $α$: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case $α\in \mathbb R$. For the asymptotics of the eigenvalues as $α\to \infty$ in $\mathbb C$, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that every Robin eigenvalue either diverges to $\infty$ in $\mathbb C$ or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where $Ω$ is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension $d\geq 2$ all eigenvalues converge to the Dirichlet spectrum if ${\rm Re}\, α$ remains bounded from below as $α\to \infty$, while if ${\rm Re}\, α\to -\infty$, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like $-α^2$.

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arXiv2019

Essential numerical ranges for linear operator pencils

We introduce concepts of essential numerical range for the linear operator pencil $λ\mapsto A-λB$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=λx$ into the pencil problem $BTx=λBx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of non-selfadjoint Schrödinger operators which it has not been possible to treat with existing methods.

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arXiv2019

The essential numerical range for unbounded linear operators

We introduce the concept of essential numerical range $W_{\!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do \emph{not} carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range $W_{\!e}(T)$ is that it captures spectral pollution in a unified and minimal way when approximating $T$ by projection methods or domain truncation methods for PDEs.

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arXiv2016

Schrödinger operator with non-zero accumulation points of complex eigenvalues

We study Schrödinger operators $H=-Δ+V$ in $L^2(Ω)$ where $Ω$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential $V\in L^p(Ω)\cap L^{\infty}(Ω)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $σ_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.

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arXiv2016

Local convergence of spectra and pseudospectra

We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local $\varepsilon$-pseudospectral exactness outside the limiting essential $\varepsilon$-near spectrum, and discuss properties of these two notions including perturbation results.

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arXiv2016

Convergence of sequences of linear operators and their spectra

We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions that ensure spectral exactness, i.e. that every true eigenvalue is approximated and no spurious eigenvalues occur. A main ingredient is the discrete compactness of the sequence of resolvents of the approximating operators. We establish sufficient conditions and perturbation results for strong convergence and for discrete compactness of the resolvents.

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arXiv2015

Approximations of spectra of Schrödinger operators with complex potentials on $\mathbb{R}^d$

We study spectral approximations of Schrödinger operators $T=-Δ+Q$ with complex potentials on $Ω=\mathbb{R}^d$, or exterior domains $Ω\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $Ω_n$, and of boundary conditions on $\partial Ω_n$ such as mixed Dirichlet/Robin type. In particular, ${\rm Re} \, Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for $d=1,2,3$, illustrate our results.

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arXiv2014

Remarks on the convergence of pseudospectra

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.

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