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Dr. rer. nat. Siegfried Beckus

Postdoktorand

Kontakt
Raum:
2.09.2.11
Telefon:
+49 331 977 2748
E-Mail:
beckus@uni-potsdam.de
...

Research interests & Preprints

Preprints

Research interests

  • spectral theory of Schrödinger operators on graphs with aperiodic ordered potentials
  • dynamical Systems
  • Delone sets
  • graph limits and graphings
  • aperiodic tilings
  • operator algebras
  • fields of C*-algebras (especially C*-algebras induced by groupoids)

PhD thesis

Spectral approximation of aperiodic Schrödinger operators
Friedrich-Schiller Universität Jena, October 2016

Diploma thesis

Generalized Bloch Theory for Quasicrystals
Friedrich-Schiller Universität Jena, February 2012

Scientific Output

@ googlescholar, mathscinet

Publications

2020 | Growth of Eigenfunctions and R-limits on Graphs | Siegfried Beckus, Latif EliazZeitschrift: Journal of Spectral TheoryLink zum Preprint

Growth of Eigenfunctions and R-limits on Graphs

Autoren: Siegfried Beckus, Latif Eliaz (2020)

Zeitschrift:
Journal of Spectral Theory

2020 | Spectral Continuity for Aperiodic Quantum Systems II. Applications of a folklore theorem | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Mathematical PhysicsReihe: 61Link zur Publikation , Link zum Preprint

Spectral Continuity for Aperiodic Quantum Systems II. Applications of a folklore theorem

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2020)

This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson–Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay–Rudin–Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.

Zeitschrift:
Journal of Mathematical Physics
Reihe:
61

2020 | Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems | Siegfried Beckus, Baptiste DevyverZeitschrift: Journal d'Analyse MathematiqueLink zum Preprint

Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems

Autoren: Siegfried Beckus, Baptiste Devyver (2020)

Zeitschrift:
Journal d'Analyse Mathematique

2019 | Hölder Continuity of the Spectra for Aperiodic Hamiltonians | Siegfried Beckus, Jean Bellissard, Horia CorneanZeitschrift: Annales Henri PoincaréReihe: 20Seiten: 3603 - 3631Link zur Publikation , Link zum Preprint

Hölder Continuity of the Spectra for Aperiodic Hamiltonians

Autoren: Siegfried Beckus, Jean Bellissard, Horia Cornean (2019)

We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.

Zeitschrift:
Annales Henri Poincaré
Reihe:
20
Seiten:
3603 - 3631

2019 | Corrigendum to “Spectral continuity for aperiodic quantum systems I. General theory | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Functional AnalysisReihe: 277Seiten: 3351-3353Link zur Publikation , Link zum Preprint

Corrigendum to “Spectral continuity for aperiodic quantum systems I. General theory

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2019)

A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.

Zeitschrift:
Journal of Functional Analysis
Reihe:
277
Seiten:
3351-3353

2018 | Shnol-type theorem for the Agmon ground state | Siegfried Beckus, Yehuda PinchoverZeitschrift: Journal of Spectral TheoryReihe: 10Seiten: 355 - 377Link zum Preprint

Shnol-type theorem for the Agmon ground state

Autoren: Siegfried Beckus, Yehuda Pinchover (2018)

Let H be a Schrödinger operator defined on a noncompact Riemannian manifold Ω, and let W∈L∞(Ω;R). Suppose that the operator H+W is critical in Ω, and let φ be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction of H satisfying |u|≤φ in Ω, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K⋐Ω the operator H admits a positive solution in Ω'=Ω∖K, and |u|≤ψ in Ω', where ψ is a positive solution of minimal growth in a neighborhood of infinity in Ω.
Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

Zeitschrift:
Journal of Spectral Theory
Reihe:
10
Seiten:
355 - 377

2018 | Spectral continuity for aperiodic quantum systems I. General theory | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Functional AnalysisReihe: 275Seiten: 2917 - 2977Link zur Publikation , Link zum Preprint

Spectral continuity for aperiodic quantum systems I. General theory

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2018)

How does the spectrum of a Schrödinger operator vary if the corresponding geometry and dynamics change? Is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures? In this work a positive answer is provided using the rather general setting of groupoid C*-algebras. A characterization of the convergence of the spectra by the convergence of the underlying structures is proved. In order to do so, the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles. With this at hand, magnetic Schrödinger operators on dynamical systems or Delone systems fall into this unified setting. Various approximations used in computational physics, like the periodic or the finite cluster approximations, are expressed through the tautological groupoid, which provides a universal model for fields of groupoids. The use of the Hausdorff topology turns out to be fundamental in understanding why and how these approximations work.

Zeitschrift:
Journal of Functional Analysis
Reihe:
275
Seiten:
2917 - 2977

2018 | Delone dynamical systems and spectral convergence | Siegfried Beckus, Felix PogorzelskiZeitschrift: Ergodic Theory and Dynamical SystemsLink zur Publikation , Link zum Preprint

Delone dynamical systems and spectral convergence

Autoren: Siegfried Beckus, Felix Pogorzelski (2018)

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

Zeitschrift:
Ergodic Theory and Dynamical Systems

2018 | Note on spectra of non-selfadjoint operators over dynamical system | Siegfried Beckus, Daniel Lenz,Marko Lindner,Christian SeifertZeitschrift: Proceedings of the Edinburgh Mathematical SocietyReihe: 61Seiten: 371 -386Link zur Publikation , Link zum Preprint

Note on spectra of non-selfadjoint operators over dynamical system

Autoren: Siegfried Beckus, Daniel Lenz,Marko Lindner,Christian Seifert (2018)

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Zeitschrift:
Proceedings of the Edinburgh Mathematical Society
Reihe:
61
Seiten:
371 -386

2017 | On the spectrum of operator families on discrete groups over minimal dynamical Systems | Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian SeifertZeitschrift: Mathematische ZeitschriftReihe: 287Seiten: 993 - 1007Link zur Publikation , Link zum Preprint

On the spectrum of operator families on discrete groups over minimal dynamical Systems

Autoren: Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian Seifert (2017)

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in ℓp(ℤ). Here, we generalize this to a large class of bounded linear operator families on Banach-space valued ℓp-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

Zeitschrift:
Mathematische Zeitschrift
Reihe:
287
Seiten:
993 - 1007

2016 | Continuity of the Spectrum of a Field of Self-Adjoint Operators | Siegfried Beckus, Jean BellissardZeitschrift: Annales Henri PoincaréReihe: 17Seiten: 3425 - 3442Link zur Publikation , Link zum Preprint

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Autoren: Siegfried Beckus, Jean Bellissard (2016)

Given a family of self-adjoint operators (At)t∈T indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum σ(At) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T,d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.

Zeitschrift:
Annales Henri Poincaré
Reihe:
17
Seiten:
3425 - 3442

2013 | Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals | Siegfried Beckus, Felix PogorzelskiZeitschrift: Mathematical Physics, Analysis and GeometryReihe: 16Seiten: 289 -308Link zur Publikation , Link zum Preprint

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

Autoren: Siegfried Beckus, Felix Pogorzelski (2013)

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.

Zeitschrift:
Mathematical Physics, Analysis and Geometry
Reihe:
16
Seiten:
289 -308

CV

CV in english, CV in deutsch

Academic career

University of Potsdam since 10/2018 Post-Doc with Prof. Dr. Matthias Keller

Israel Institute of Technology (Technion), Haifa 10/2016-09/2018 Postdoctoral Fellowship with Prof. Dr. Yehuda Pinchover and Prof. Dr. Ram Band

Georgia Institute of Technology, Atlanta, USA 02-04/2014 Visiting Student/Research Collaborator

Friedrich Schiller University Jena 03/2012-09/2016 PhD Student

Education

PhD 10/2016, Friedrich Schiller University Jena

Diploma 02/2012, Friedrich Schiller University Jena

Grants and Projects

Marie-Weber-Grant 2022 "Aperiodische Ordnung: Die Entschlüsselung versteckter Strukturen" starting 09/2022 (PhD position for 1 year) - announcement by the Hans Böckler Stiftung

DFG Project "Periodic approximations of Schrödinger operators associated with quasicrystals" since 12/2018 (PostDoc position for 2 year and travel money)

ZiF Summer Camp 2020 Research stay at the Center of Interdisciplinary Research at the University Bielefeld, Bielefeld, Germany

Scholarship of the DAAD Program to participate at congresses, International Workshop on Operator Theory and its Applications, Lisbon, Portugal

Scholarship for "Research in Pairs" at the Mathematisches Forschungsinstitut Oberwolfach, Germany, 01/2018

Postdoctoral Fellowship Israel Institute of Technology (Technion), Haifa, Israel, 10/2016 - 09/2018

Scholarship of the DAAD Program to participate at congresses, XVIII International Congress on Mathematical Physics, Santiago de Chile, Chile

Funding for the PhD Seminar "Förderung von interdisziplinären Arbeitsgruppen und Nachwuchsnetzwerken" funded by the Graduierten-Akademie in Jena

(Co-)Supervision

Lior Tennenbaum (PhD thesis), joint with Prof. Ram Band (Israel Institute of Technology)

Alberto Takase (PhD thesis, joint with Prof. Anton Gorodetski (University of California Irvine)

Franziska Sieron (Master thesis), The density of periodic configurations in strongly irreducible subshifts of finite type (joint with Prof. Dr. Daniel Lenz), 2016 

Daniel Sell (Master thesis), Topological groupoids and Matuis spatial realization theorem (joint with Prof. Dr. Daniel Lenz), 2015

Franziska Sieron (Bachelor thesis), The balanced property of primitive substitutions (joint with Prof. Dr. Daniel Lenz), 2014

(Co-)Organized Scientific meetings

Summer school on "Spectral theory and geometry of ergodic Schrödinger operators", Universität Potsdam, 07/2023-08/2023, Link

Workshop on "Functional Analysis, Operator Theory and Dynamical Systems",  Universität Potsdam, 09/2022, Link

Euler-Lecture, Universität Potsdam, 05/2019, 05/2021, 05/2022, 05/2023, Link

PhD seminar, Friedrich-Schiller-Universität Jena, 03/2013 – 09/2016

Colloquium: "Job opportunities for mathematicians", Friedrich-Schiller-Universität Jena, 2013 – 2015

PhD symposium at the TU Chemnitz, within the Fall school Dirichlet forms, operator theory and mathematical physics, 02/2013

Selected Talks at international conferences

  • 06/2022 Workshop "Ergodic Operators and Quantum Graphs", Simons Center for Geometry and Physics (USA)
    Spectral approximations beyond dimension one
  • 04/2022 Workshop "Almost-Periodic Spectral Problems", Banff International Research Station (Canada)
    The table and the chair: Spectral approximations beyond dimension one Link to the video
  • 01/2021 Workshop "Geometry, Dynamics and Spectrum of Operators on Discrete Spaces", MFO Oberwolfach (Germany):
    Primitive substitutions beyond abelian groups: The Heisenberg group
  • 07/2019 International Workshop on Operator Theory and its Applications (IWOTA 2019), Instituto Superior Técnico, Lisbon (Portugal):
    Hunting the spectra via the underlying dynamics
  • 05/2019 8th Miniworkshop on Operator Theoretic Aspects of Ergodic Theory, Leipzig (Germany):
    Hunting the spectra via the underlying dynamics
  • 01/2018 Hardy-type inequalities and elliptic PDEs, Midreshet Sde Boker (Israel), Poster:
    Spectral Approximation of Schrödinger Operators
  • 10/2017 Workshop "Spectral Structures and Topological Methods in Mathematical Quasicrystals", MFO Oberwolfach (Germany):
    Spectral stability of Schrödinger operators in the Hausdorff metric
  • 07/2017 Analysis and Geometry on Graphs and Manifolds, Universität-Potsdam (Germany):
    Shnol type Theorem for the Agmon ground state
  • 05/2017 Israel Mathematical Union 2017, Acre (Israel):
    The space of Delone dynamical systems and related objects
  • 01/2017 Workshop on Mathematical Physics, Weizmann Institute of Science, Rehovot (Israel), Poster:
    Spectral Approximation of Schrödinger Operators
  • 06/2016 Thematic School "Transversal Aspects of Tilings", Oleron (France):
    Continuity of the spectra associated with Schrödinger operators
  • 09/2015 CMO-BIRS, Workshop on "Spectral properties of quasicrystals via analysis, dynamics and geometric measure theory", Oaxaca (Mexiko):
    Spectral approximation of Schrödinger operators: continuity of the spectrum
  • 07/2015 Young researcher symposium, Pontificia Universidad Catolica de Chile, Santiago de Chile (Chile):
    Spectral study of Schrödinger operators with aperiodic ordered potential in one-dimensional systems
  • 06/2015 Workshop on "Time-frequency analysis and aperiodic order", Norwegian University of Science and Technology, Trondheim (Norway):
    An approximation theorem for the spectrum of Schrödinger operators related to quasicrystals

Selected Talks in seminars and colloquia

  • 04/2022 University of California - Irvine (USA):
    Sturmian dynamical systems and the Kohmoto butterfly
  • 01/2020 Universität Leipzig (Germany):
    When do the spectra of self-adjoint operators converge?
  • 05/2019 Justus-Liebig-Universität Gießen (Germany):
    Hunting the spectra via the underlying dynamics
  • 07/2018 Technische Universität München (Germany):
    When do the spectra of self-adjoint operators converge?
  • 10/2017 Pontificia Universidad Catolica de Chile, Santiago (Chile):
    Shnol type Theorem for the Agmon ground state
  • 10/2017 Hebrew University of Jerusalem (Israel):
    When do the spectra of self-adjoint operators converge?
  • 08/2017 University of Oslo (Norway):
    Spectral approximation via an approach from C*-algebras
  • 07/2017 Friedrich-Alexander Universität Erlangen-Nürnberg (Germany):
    The space of Delone dynamical systems and its application
  • 07/2017 RWTH Aachen (Germany):
    Shnol type Theorem for the Agmon ground state
  • 09/2016 Aalborg University (Danemark):
    Continuous variation of the spectra: A characterization and a tool
  • 07/2016 Universität Bielefeld (Germany):
    Hölder-continuous behavior of the spectra associated with self-adjoint operators
  • 05/2015 Technische Universität Chemnitz (Germany):
    Schrödinger operators on quasicrystals
  • 04/2015 Israel Institute of Technology (Technion), Haifa (Israel):
    The role of Gähler-Anderson-Putnam graphs in the view of Schrödinger operators
  • 04/2014 University of Alabama at Birmingham (USA):
    Gähler-Anderson-Putnam graphs of 1-dimensional Delone sets of finite local complexity
  • 01/2014 Technische Universität Hamburg-Harburg (Germany):
    Wannier transformation for Schrödinger operators with aperiodic potential

Teaching

Summer semester 2023

Autumn semester 2022

  • Exercise: Mathematik für Physik 3
  • Lecture and Exercise: Functional Analysis 1

Summer semester 2022

  • Exercise: Mathematik für Physik 2
  • Seminar: Random operators

Autumn semester 2021

  • Exercise: Mathematik für Physik 1
  • Exercise and Tutorial: Mathematik für Wirtschaftsinformatiker
  • Seminar: Aperiodische Ordnung / Aperiodic Order

Summer semester 2021

Autumn semester 2020

Summer semester 2020

Autumn semester 2019

Summer semester 2019

Autumn semester 2018

Anschrift

Universität Potsdam
Institut für Mathematik
Campus Golm, Haus 9
Karl-Liebknecht-Straße 24-25
14476 Potsdam