arXiv subjects

Albrecht Seelmann

Publications and source records attributed to Albrecht Seelmann.

27 records · Page 2Linked to original sources

arXiv2020

Band edge localization beyond regular Floquet eigenvalues

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

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arXiv2019

Semidefinite perturbations in the subspace perturbation problem

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan $ \sin2Θ$ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] adapted to this situation is proved. Under a certain additional geometric assumption on the separation of the spectrum of the unperturbed operator, this leads to a sharp estimate on the norm of the difference of the spectral projections associated with isolated components of the spectrum of the perturbed and unperturbed operators, respectively. Without this additional geometric assumption on the isolated components of the spectrum of the unperturbed operator, a corresponding estimate is obtained by transferring the optimization approach for general perturbations in [J. Anal. Math. 135 (2018), 313--343] to the present situation.

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arXiv2018

Exhaustion approximation for the control problem of the heat or Schrödinger semigroup on unbounded domains

We consider the control problem of the heat equation on bounded and unbounded domains, and more generally the corresponding inhomogeneous equation for the Schrödinger semigroup. We show that if the sequence of null-controls associated to an exhaustion of an unbounded domain converges, then the solutions do in the same way, and that the control cost estimate carries over to the limiting problem on the unbounded domain. This allows to infer the controllability on unbounded domains by studying the control problem on a sequence of bounded domains.

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arXiv2016

On invariant graph subspaces

In this paper we discuss the problem of decomposition for unbounded $2\times 2$ operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.

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arXiv2015

Notes on the subspace perturbation problem for off-diagonal perturbations

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded off-diagonal perturbation is studied. To this end, the optimization approach for general perturbations in [J. Anal. Math., to appear; arXiv:1310.4360 (2013)] is adapted. It is shown that, in contrast to the case of general perturbations, the corresponding optimization problem can not be reduced to a finite-dimensional problem. A suitable choice of the involved parameters provides an upper bound for the solution of the optimization problem. In particular, this yields a rotation bound on the subspaces that is stronger than the previously known one from [J. Reine Angew. Math. (2013), DOI:10.1515/crelle-2013-0099].

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arXiv2013

On an estimate in the subspace perturbation problem

The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. In the approach presented here, a constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate on the arcsine of the norm of the difference of the corresponding spectral projections. The problem is solved explicitly. This optimizes the approach by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. In particular, the resulting estimate is stronger than the one obtained there.

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arXiv2013

Notes on the sin2Θ theorem

An analogue of the Davis-Kahan sin2Θ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] is proved under a general spectral separation condition. This extends the generic sin2θ estimates recently shown by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively.

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arXiv2013

Reducing graph subspaces and strong solutions to operator Riccati equations

The problem of block diagonalization for diagonally dominant symmetric block operator matrices with self-adjoint diagonal entries is considered. We show that a reasonable block diagonalization with respect to a reducing graph subspace requires a related skew-symmetric operator to be a strong solution to the associated Riccati equation. Under mild additional regularity conditions, we also establish that this skew-symmetric operator is a strong solution to the Riccati equation if and only if the graph subspace is reducing for the given operator matrix. These regularity conditions are shown to be automatically fulfilled whenever the corresponding relative bound of the off-diagonal part is sufficiently small. This extends the results by Albeverio, Makarov, and Motovilov in [Canad. J. Math. Vol. \textbf{55}, 2003, 449--503], where the off-diagonal part is required to be bounded.

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arXiv2010

Metric properties of the set of orthogonal projections and their applications to operator perturbation theory

We prove that the set of orthogonal projections on a Hilbert space equipped with the length metric is $\frac\pi2$-geodesic. As an application, we consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators and obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unpertubed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476] are sharpened.

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