# Research

Research interests and funding - Daniel Lenz
• Geometry and spectral theory of Dirichlet spaces

• Random and equivariant Schrödinger operators

• Aperiodic order and dynamical systems of low complexity

Research supported by German Research Foundation (DFG) in Einzelförderung ' Green's formulae and harmonic functions for graphs and Dirichlet spaces - follow up' (Joint with Prof. Dr. Matthias Keller and Dr. Marcel Schmidt), starting 2018.

Research supported by German Research Foundation (DFG) within Schwerpunktprogramm 'Geometry at infinity', since spring 2017.

Research supported by German Research Foundation (DFG) within Graduiertenkolleg 'Quanten- und Gravitationsfelder' since  fall 2013.

Research supported  by German Research Founddation (DFG) in Einzelförderung 'Geometry of discrete spaces and spectral theory of non-local operators' (joint with Dr. Matthias Keller),   since 2012.

Research supported by German Research Foundation (DFG) in Einzelföderung 'Zuf\"allige und periodische Quantengraphen', 2009 - 2014.

Research supported by German Research Foundation (DFG) in Einzelförderung  'Delone dynamische Systeme: Ergodentheorie, Diffraktion und Operatortheorie'    (joint project with Prof. Dr. P. Stollmann), 2004 --2008.

Research supported by  German Research Foundation (DFG) within Schwerpunktprogramm "Quasikristalle"  (joint project with Prof. Dr. P. Stollmann), 2001 - 2003.

Publications - Daniel Lenz
• Blow-up of nonnegative solution of an abstract  semilinear heat equation with convex source (with Marcel Schmidt, Ian Zimmermann), (2108.11291)
• On L^p Liouville theorems for Dirichlet forms (with Bobo Hua, Matthias Keller, Marcel Schmidt), (2108.11678

• Pure point spectrum for dynamical systems and mean almost periodicity (with Timo Spindeler, Nicolae Strungaru),  (2006.10825)

• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity  (with Timo Spindeler, Nicolae Strungaru), (2006.10821)

• Self-Adjoint Extensions of Bipartite Hamiltonians (with Timon Weinmann, Melchior Wirth), to appear in: Proc. Edinburgh Math. Soc. (1912.03670)

• Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schrödinger Operators on the Real Line  (with David Damanik), to appear in: Israel J. Math. (1909.00378)

• Modulated crystals and almost periodic measures (with Jeong-Yup Lee, Christoph Richard, Bernd Sing, Nicolae Strungaru) to appear in: Lett. Math. Phys. (1907.07017)

• Subshifts with leading sequences, uniformity of cocycles and spectral theory of Schreier graphs (with Rostislav Grigorchuk, Tatiana Nagnibeda, Daniel Sell). (1906.01898)
• Universal lower bounds for Laplacians on weighted graphs (with Peter Stollmann), to appear in: Analysis and Geometry on Graphs and Manifolds, eds: Matthias Keller, Daniel Lenz, Radoslaw Wojciechowski, London Mathematical Society Lecture Note Series 461.  (1903.02382)

• The structure of mean equicontinuous group Actions (with Gabriel Fuhrmann, Maik Gröger), to appear in: Israel J. Math. (1812.10219)

• Spectral theory of dynamical systems as diffraction theory of sampling functions, to appear in:  Monatshefte f. Mathematik  (1809.07639)

• Topological Poincare type  inequalities and lower bounds on the infimum of the spectrum for graphs (with Marcel Schmidt, Peter Stollmann). (1801.09279)

• Geometric properties of Dirichlet forms under order isomorphism (with Marcel Schmidt, Melchior Wirth). (1801.08326)

• Combinatorics of the subshift associated to Grigorchuk's group (with Rostislav Grigorchuk, Tatiana Nagnibeda),  (Russian) Tr. Mat. Inst. Steklova   297   (2017), Poryadok i Khaos v Dinamicheskikh Sistemakh, 158--164.  (1711.10342)

• Boundary representation of Dirichlet forms on discrete spaces (with Matthias Keller, Marcel Schmidt, Michael Schwarz), J. Math. Pures App. 126 (2019), 109--143.  (1711.08304)

• Domination of quadratic forms (with Marcel Schmidt, Melchior Wirth), to appear in: Mathematische Zeitschrift  (1711.07225)

• On the decomposition principle and a Persson type theorem for general regular Dirichlet forms (with Peter Stollmann),   J.  Spectr.  Theory  9   (2019),   1089--1113.. (1705.10398)

• Averaging almost periodic functions along exponential sequences (with Michael Baake, Alan Haynes),  in: Aperiodic order: Volume 2, Crystallography and Almost Periodicity.  (Eds M. Baake, U. Grimm), Cambridge University Press, (2017).   (1704.08120)

• Diffusion on Delone sets (with Sebastian Haeseler, Xuping Huang, Felix Pogorzelski),  J. Stat. Phys. 167  (2017),  1496--1510.  (1611.01059).

• On weakly almost periodic measures (with Nicolae Strungaru),  Trans. Amer. Math. Soc.  371 (2019), 6843--6881.  (1609.08219)

• Uniqueness of form extensions and domination of semigroups (with Melchior Wirth, Marcel Schmidt), to appear in: J. Funct. Anal.  (1608.06798)

• An autocorrelation and discrete spectrum for dynamical systems on metric spaces, to appear in: Ergodic Theory & Dynamical Systems  (1608.05636)

• On the spectrum of operator families on discrete groups over minimal dynamical systems (with Siegfried Beckus, Marko Lindnder, Christian Seifert),   Math. Z.  287     (2017),    993--1007.  (1606.08353)

• An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs (with Peter Stollmann, Guenter Stolz), to appear in: J. Spectral Theory.   (1606.07476)

• Model sets with positive entropy in Euclidean cut and project schemes (with Tobias Jaeger, Christian Oertel), to appear in: Ann.  sc. ENS.  (1605.01167)

• Spectral notions of aperiodic order (with Michael Baake),  Discrete Contin. Dyn. Syst. Ser. S  10  (2017),  161--190. (1601.06629)

• Toeplitz flows and model sets (with Michael Baake, Tobias Jaeger), Bull. London Math. Soc. 48 (2016), 691--698. (1511.08595)

• Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive Substitution (with Rostislav Grigorchuk,  Tatiana Nagnibeda), to appear in: Groups, Graphs and Random Walks, T. Ceccherini-Silberstein, M. Salvatori, E. Sava-Huss, (Eds), London Math. Soc. Lecture Note Series, Cambridge  University Press  (2017).  (1510.00545)

• Note on short time behavior of semigroups associated to selfadjoint Operators (with Matthias Keller, Florentin Münch, Marcel Schmidt, Andras Telcs),  Bull. London Math. Soc. 48 (2016), 935--944.  (1509.01993)

• Invariant means on Boolean inverse monoids (with Ganna Kudryavtseva, Mark Lawson, Pedro Resende),  Semigroup Forum 92 (2016), 77--101.  (1503.03733)

• Note on the set of Bragg peaks with high intensity (with Nicolae Strungaru),  Ann. Henri Poincare 17 (2016), 673--687. (1412.7377)
• Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order (with Rostislav Grigorchuk, Tatiana Nagnibeda),   Math. Ann.   370 (2018), 1607 --1637.    (1412.6822)

• Note on spectra of non-selfadjoint operators over dynamcial systems (with Siegfried Beckus, Marko Lindnder, Christian Seifert),  Proc. Edinburgh Math. Soc.  61 (2018), 371--386.   (1412.5926)

• Global properties of Dirichlet forms in terms of Green's formula (with Sebastian Haeseler, Matthias Keller, Jun Masamune, Marcel Schmidt),   Calc. Var. PDE 56  (2017),   43 pp.  (1412.3355)

• Note on uniformly transient graphs (with Matthias Keller, Marcel Schmidt, Radek Wojciechowski), Rev. Mat. Iberoam.  33  (2017),   831--860.  (1412.0815)

• The Ihara Zeta function for infinite graphs (with Felix Pogorzelski, Marcel Schmidt),  Trans. Amer. Math. Soc. 371 (2019), 5687--5729.   (1408.3522)

• Equicontinuous factors, Proximality and Ellis semigroup for Delone sets (with Jean-Baptiste Aujogue, Marcy Barge, Johannes Kellendonk), in:  Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz, J. Savinien), Progress in Mathematics 309, Birkhaeuser,  (1407.1787)

• Diffusion determines the recurrent graph (with Matthias Keller, Marcel Schmidt, Melchior Wirth),  Advances in Mathematics 269 (2015), 364--398.  (1405.3256)

• An invitation to trees of finite cone type: random and deterministic Operators (with Matthias Keller, Simone Warzel), Markov Processes Relat. Fields 21 (2015), 557--574.     (1403.4426)

• Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces (with Alexander Teplyaev), Trans. Amer. Math. Soc.   368 (2016),  4933--4956.  (1310.5650)

• Graphs of finite measure (with Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Radek Wojciechowski),   J. Math. Pures Appl.   103 (2015), 1093--1131.  (1309.3501)
• Zero measure Cantor spectra for continuum one-dimensional quasicrystals (with Christian Seifert, Peter Stollmann), J. Diff. Eqn. 256 (2014), 1905--1926. (1308.3431)
• Dynamical versus diffraction spectrum for structures with finite local complexity (with Michael Baake, Aernout van Enter),  Ergodic Theory & Dynamical Systems   35  (2015),  2017--241.  (1307.7518)
• New relations between discrete and continuous transition operators on (metric) graphs (with Konstantin Pankrashkin),  Integral Equations and Operator Theory 84 (2016), 151--181.   (1305.7491)

• Distributive inverse semigroups and non-commutative Stone dualities (with Mark Lawson), (1302.3032)

• Unbounded quantum graphs with unbounded boundary conditions (with Carsten Schubert and Ivan Veselic),  Math. Nachr.  287 (2014), 962--979.  (1205.1944)

• Delone sets  with finite local complexity: Linear repetitivity versus positivity of weights (with Adnene Besbes, Michael Boshernitzan), Discrete and Computational Geometry   49 (2013), 335--347. (1202.6023)
• Stationary processes with pure point diffraction (with Robert V. Moody),  Ergodic Theory & Dynamical Systems  37  (2017),  2597--2642.    (1111.3617)

• A characterization of subshifts with bounded powers (with Johannes Kellendonk and Jean Savinien),  Disc. Math. 313 (2013), 2881--2894.  (1111.1609)

• Absolutely continuous spectrum for random operators on trees of finite cone type (with Matthias Keller and Simone Warzel),  Journal d' Analyse Mathematique  118 (2012), 363---396.   (1108.0057)

• Pseudogroups and their etale groupoids (with Mark V. Lawson), Advances in Mathematics 244 (2013), 117--170.  (1107.5511)

• Equicontinuous Delone dynamical systems (with Johannes Kellendonk), Canadian Journal of Mathematics 65 (2013), 149--170.    (1105.3855).

• Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs (with Matthias Keller, Rados{\l}aw Wojciechoswki), Mathematische Zeitschrift  274 (2013), 905--932.   (1105.0395).

• Laplacians on infinite graphs: Dirichlet and Neumann  boundary conditions (with Sebastian Haeseler, Matthias Keller, Rados{\l}aw Wojciechowski),   Journal of Spectral Theory  2 (2012), 397--432.   (1103.3695).

• Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation (with Matthias Keller), Math. Model. Nat. Phenom. 5 (2010), 198--224.  (1101.2979)

• Note on basic features of large time behaviour of heat kernels (with Matthias Keller, Hendrik, Vogt,  Rados{\l}aw Wojciechowski),  J. reine  angew. Math. (Crelles Journal) 708 (2015),  73--95.   (1101.0373)
• Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory (with Rupert Frank, Daniel Wingert), J. Funct. Anal. 266 (2014),  4765--4808.  (1012.5050)

• L^p approximation of the integrated density of states for Schrödinger operators with finite local complexity (with Michael Gruber, Ivan Veselic), Integr. Equ. Oper. Theory  69, (2011), 217--232.  (1004.3471)

• A Banach space valued ergodic theorem and uniform approximation of the integrated density of states (with Fabian Schwarzenberger, Ivan Veselic),  Geometriae Dedicata 150 (2011), 1--34.   (1003.3620), Erratum: Geometriae Dedicata 159 (2012), 411--413.

• Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals  (with Steffen Klassert, Peter Stollmann),  Discrete and Continuous Dynamical Systems - Series  A  29 (2011),  1553--1571. (1003.3574)

• On the spectral theory of trees with finite cone type (with Matthias Keller, Simone Warzel),  Israel Journal of Mathematics  194 (2013),  107--135. (1001.3600)

• Note on powers in three interval exchange transformations (with Zuzana Masakova, Edita Pelantova),   Theoret. Comput. Sci. 412 (2011), 3788--3794.    (0909.1109)

• Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms (with  Peter Stollmann, Ivan Veselic),  Oper. Theory Adv. Appl. 214 (2011),  83--106.  (0909.1107)

• An uncertainty principle, Wegner estimates and localization near fluctuation boundaries (with Anne Boutet de Monvel, Peter Stollmann),  Math. Z. 269 (2011), 663--670.   (0905.2845)

• Dirichlet forms and stochastic completeness of graphs and subgraphs (with Matthias Keller), J. reine angew. Math. (Crelles Journal) 666 (2012), 189--223.   (0904.2985)

• Compactness of Schrodinger semigroups (with Peter Stollmann, Daniel Wingert),  Math. Nachrichten 283 (2010), 94--103. (0903.0280)

• Mini-Workshop: The Pisot Conjecture - from Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets. Abstracts from the mini-workshop held March 1-7, 2009. Organized by Valerie Berthe, David Damanik and Daniel Lenz. Oberwolfach Reports. Vol 6. no 1.  (2009),  725--766.

• Continuity of the integrated density of states on random length metric graphs (with Norbert Peyerimhoff, Olaf Post, Ivan Veselic'),  Mathematical Physics, Analysis and Geometry  12 (2009), 219--254.  (0811.4513)

• The Allegretto-Piepenbrink Theorem for strongly local forms (with Peter Stollmann, Ivan Veselic),  Documenta Mathematica  14 (2009), 167--189.   (0811.2135)

• Extinctions and correlations for uniformly discrete point processes with pure point spectrum (with Robert V. Moody),   Commun. Math. Phys. 289, (2009), 907--923.  (0902.0567)

• Aperiodic order and pure point diffraction, The Philosophical Magazine, 88 (2008), 2059-2071 (Special Issue: Quasicrystals: the silver jubilee) (0802.324)

• Eigenfunction expansion for Schrodinger operators on metric graphs (with Carsten Schubert, Peter Stollmann), Integral Equations and Operator Theory 62 (2008), 541--553. (0801.1376)

• Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d (with Michael Gruber, Ivan Veselic), in: Analysis on Graphs and its Applications (edited by P. Exner, J. Keating, P.Kuchment T. Sunada and A. Teplyaev), Proceedings of Symposia in Pure Mathematics 77 (0712.1740)

• Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity, Contemp. Math. 485 (Ergodic Theory, Idris Assani, ed), (2009), 91 --112. (0712.1323)

• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence (with Ivan Veselic),   Mathematische Zeitschrift 263 (2009), 813--835.   (0709.2836)

• Sch'nol's Theorem For Strongly Local Forms (with Anne Boutet de Monvel, Peter Stollmann), Israel Journal of Mathematics 173 (2009), 189--211. (0708.1501)

• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies (with Michael Baake, Christoph Richard), Lett. Math. Phys. Volume 82, (2007), 61--77. (0706.1677)

• Continuity properties of the integrated density of states on manifolds (with Norbert Peyerimhoff, Olaf Post, Ivan Veselic), Japanese Journal of Mathematics, 3 (2008), 121--161. (0705.1079)

• Pure point spectrum for measure dyamical systems on locally compact Abelian groups (with Nicolae Strungaru),  J. Math. Pures Appl. 92 (2009), 323--341.  (0704.2498)

• Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over $\mathbb{Z}^d$ (with Michael J. Gruber, Ivan Veselic), J. Funct. Anal. Volume 253, (2007), 515--533. (math.0612743)

• Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys. 287 (2009), 225--258. (math-ph/0608026)

• Uniform existence of the integrated density of states for models on $\ZZ^d$ (with Peter Mueller and Ivan Veselic), Positivity 12 (2008), 571 - 589. (math-ph/0607063)

• Mini-workshop: $L\sp 2$-Spectral Invariants and the Integrated Density of States. Abstracts from the mini-workshop held February 19--25, 2006. Organized by Jozef Dodziuk, Daniel Lenz, Thomas Schick and Ivan Veseli\'c. Oberwolfach Reports. Vol. 3, no. 1. (2006), no. 1, 511--552.

• Pure point diffraction and cut and project schemes for measures: The smooth case (with Christoph Richard), Mathematische Zeitschrift 256, (2007), 347--378. (math.DS/0603453)

• On an order based construction of a groupoid from an inverse semigroup, Proc. Edinburgh Math. Soc. 51 (2008), 387--406. (math.OA/0604105)

• Uniform Szego Cocycles Over Strictly Ergodic Subshifts (with David Damanik), Journal of approximation theory 144, (2007), 133-138. (math.SP/0512033)

• Characterizations of model sets by dynamical systems (with Michael Baake, Robert V. Moody), Ergodic Theory & Dynamical Systems 27, (2007), 341-382. (math.DS/0511648)

• Mini-Workshop: Dynamics of Cocycles and One-Dimensional Spectral Theory. Abstracts from the mini-workshop held November 13--19, 2005. Organized by David Damanik, Russell Johnson and Daniel Lenz. Oberwolfach Reports. Vol. 2, no. 4. (2005), 2933--2978.

• Generic subsets in spaces of measures and singular continuous spectrum (with Peter Stollmann), in: Lecture Notes in Physics 690, Springer (2006)

• Lower Transport Bounds for One-Dimensional Continuum Schr\"odinger Operators (with David Damanik, Guenter Stolz ), Math. Ann. 336, (2006), 361--389 (math-ph/0410062 )

• Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature (with Steffen Klassert, Norbert Peyerimhoff, Peter Stollmann), Proceedings of the American Mathematical Society, 134, (2006) 1549--1559. (math-ph/0410022)

• Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians (with Peter Stollmann), Duke Math. J. 131, (2006), 203--217. (math-ph/0410021)

• Zero-measure Cantor spectrum for Schrodinger operators with low-complexity potentials (with David Damanik) J. Math. Pures Appl. (9) 85 (2006), 671--686

• Deformation of Delone dynamical systems and pure point spectrum (with Michael Baake), Journal of Fourier Analysis and Applications, 11, (2005), 125 --150 (math.DS/0404155)

• A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem (with David Damanik), Duke Math. J. 133 (2006), 95--123

• An ergodic theorem for Delone dynamical systems and existence of the integrated density of states (with Peter Stollmann), J. d' Analyse Mathematique 97, (2006), 1 --23. (math-ph/0310017)

• Singular continuous spectrum for certain quasicrystal Schrödinger operators, Contemporary Mathematics 364, 169--180

• Substitution dynamical systems: Characterization of linear repetitivity and applications (with David Damanik), J. Math. Anal. Appl. 321 (2006), 766--780 (math.DS/0302231)

• Aperiodic order and quasicrystals: spectral properties, Annales Henri Poincare 4, suppl 2 (2004). (with Peter Stollmann)

• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra (with Michael Baake), Ergodic Theory & Dynamical Systems 24 (2004), 1867-1893. (math.DS/0302061)

• Ergodic theory and discrete one-dimensional random Schr"odinger operators: uniform existence of the Lyapunov exponent, Contemporary Mathematics 327 (2003), 223--238.

• Integrated density of states for random metrics on manifolds (with Norbert Peyerimhoff and Ivan Veselic), Proc. London Math. Soc. 88 (2004), 733--752. (math-ph/0212058)

• Random Schr"odinger operators on manifolds (with Norbert Peyerimhoff and Ivan Veselic), Markov Process. Related Fields 9 (2003), 717--728, proceedings of the conference "Aspects Mathematiques des Systemes Aleatoires et de la Mecanique Statistique", held in the honour of L. Pastur, May 2002, Marseille (math-ph/0212057)

• Delone dynamical systems: ergodic features and applications (with Steffen Klassert and Peter Stollmann), in: Quasicrystals, Structure and Physical Properties, H.-R. Trebin (ed.), Wiley-VCH, Berlin (2003), 172--187.

• Algebras of random operators associated to Delone dynamical systems (with Peter Stollmann), Mathematical Physics, Analysis and Geometry 6 (2003), 269--290. (math-ph/0210031)

• Aperiodic linearly repetitive Delone sets are densely repetitive, Discrete Comput. Geom. 31 (2004), 323--326. (math.MG/0208132)

• Discontinuities of the integrated density of states for random operators on Delone sets (with Steffen Klassert and Peter Stollmann), Commun. Math. Phys. 241 (2003), 235--243. (math-ph/0208027)

• Existence of non-uniform cocycles on uniquely ergodic systems, Ann. Inst. Henri Poincare: Prob. & Stat. 40 (2004), 197--206.

• Powers in Sturmian sequences (with David Damanik), European J. Combin. 24 (2003), 377--390.

• Groupoids, von Neumann Algebras and the Integrated Density of States (with Norbert Peyerimhoff and Ivan Veselic), Math. Phys. Anal. Geom. 10 (2007), 1--41. (arXiv:math-ph/0203026)

• Delone dynamical systems and associated random operators (with Peter Stollmann), Conference Proceedings, Constanta (Romania), July 2-7, 2001, J.-M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop, S. Stratila (eds.), Theta Foundation, ISBN 973-85432-2-3. (arXiv:math-ph/0202042)

• Quasicrystals, aperiodic order, and groupoid von Neumann algebras (with Peter Stollmann), C. R. Acad. Sci. Ser. I, 334 (2002), 1131--1136.

• Uniform spectral properties of one-dimensional quasicrystals, IV. quasi-Sturmian potentials (with David Damanik), J. d' Analyse Mathematique 90 (2003), 115--139. (arXiv:math-ph/0105034)

• Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals, Commun. Math. Phys., 227 (2002), 119--130. (arXiv:math-ph/0106012)

• Uniform ergodic theorems on subshifts over a finite alphabet, Ergodic Theory & Dynamical Systems, 22 (2002), 245--255. (arXiv:math.DS/0005067)

• The index of Sturmian sequences (with David Damanik), European Journal of Combinatorics, 23 (2002), 23--29.

• Linear repetitivity I., Uniform subadditive ergodic theorems and applications (with David Damanik), Discrete and Computational Geometry, 26 (2001), 411-428. (arXiv:math.DS/0005062)

• Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue mesasure (with David Damanik), Forum Mathematicum 16 (2004), 109--128 (arXiv:math.SP/9905099)

• Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-continuity (with David Damanik and Rowan Killip), Commun. Math. Phys. 212 (2000), 191-204. (arXiv:math-ph/9910017)

• Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent (with David Damanik), Lett. Math, Phys. 50 (1999), 245-257. (arXiv:math-ph/9905008)

• Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues (with David Damanik), Commun. Math. Phys. 207 (1999), 687-696. (arXiv:math-ph/9903011)

• Hierarchical structures in Sturmian dynamical Systems, preprint 1999, revised version 2001, Theoret. Comput. Sci. 303 (2003).

• Random Operators and Crossed Products, Mathematical Physics, Analysis and Geometry, 2 (1999), 197-220. (arXiv:math-ph/9902030)
Publications - Group

## 2019

• Self-Adjoint Extensions of Bipartite Hamiltonians  (1912.03670)
Daniel Lenz, Timon Weinmann, Melchior Wirth
• Modulated crystals and almost periodic measures (1907.07017)
Jeong-Yup Lee, Daniel Lenz,  Christoph Richard, Bernd Sing, Nicolae Strungaru
• Subshifts with leading sequences, uniformity of cocycles and spectral theory of Schreier graphs (1906.01898)
Rostislav Grigorchuk, Daniel Lenz,  Tatiana Nagnibeda, Daniel Sell
• Universal lower bounds for Laplacians on weighted Graphs  (1903.02382)
Daniel Lenz, Peter Stollman

## 2018

• The structure of mean equicontinuous group Actions (1812.10219)
Gabriel Fuhrmann, Maik Gröger, Daniel Lenz , 33pp.
• Spectral theory of dynamical systems as diffraction theory of sampling functions  (1809.07639)
Daniel Lenz, 23 pp.
• Topological Poincaré type inequalities and lower bounds on the infimum of the spectrum for graphs (arXiv:1801.09279)
Daniel Lenz, Marcel Schmidt, Peter Stollmann, 23pp.
• Combinatorics of One-Dimensional Simple Toeplitz Subshifts (arXiv:1801.08778)
Daniel Sell, 37pp.
• Geometric properties of Dirichlet forms under order isomorphisms (arXiv:1801.08326)
Daniel Lenz, Melchior Wirth, Marcel Schmidt, 30pp.

## 2017

• Combinatorics of the subshift associated to Grigorchuk's group (arXiv:1711.10342, journal, english translation)
Rostislav Grigorchuk, Daniel Lenz, Tatiana Nagnibeda,  Tr. Mat. Inst. Steklova 297 (2017), Poryadok i Khaos v Dinamicheskikh Sistemakh, 158-164 (in russian). english translation: Proceedings of the Steklov Institute of Mathematics, 297(1): 138-144, 2017.
• Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles (arXiv:1711.08903)
Christian Richter, Melchior Wirth, 11pp.
• A note on reflected Dirichlet forms (arXiv:1711.08746)
Marcel Schmidt, 35pp.
• Boundary representation of Dirichlet forms on discrete spaces (arXiv:1711.08304)
Daniel Lenz, Matthias Keller, Marcel Schmidt, Michael Schwarz, 38pp.
• Domination of quadratic forms (arXiv:1711.07225)
Daniel Lenz, Marcel Schmidt, Melchior Wirth, 24pp.
• Stability of Kac regularity under domination of quadratic forms (arXiv:1709.04164)
Melchior Wirth, 6pp.
• On the decomposition principle and a Persson type theorem for general regular Dirichlet forms (arXiv:1705.10398)
Daniel Lenz, Peter Stollmann, 19pp.
• Averaging almost periodic functions along exponential sequences (arXiv:1704.08120)
Michael Baake, Alan Haynes, Daniel Lenz, 18pp.
• Energy forms, PhD Thesis (arXiv:1703.04883)
Marcel Schmidt, 196pp.

## 2016

• Diffusion on Delone sets (arXiv:1611.01059)
Sebastian Haeseler, Xueping Huang, Daniel Lenz, Felix Pogorzelski, 15pp.
• Spectral approximation of aperiodic Schrödinger Operators, PhD thesis (arXiv:1610.05894)
Siegfried Beckus, 244pp.
• On Weakly Almost Periodic Measures (arXiv:1609.08219)
Daniel Lenz, Nicolae Strungaru, 43pp.
• Bakry-Emery curvature and diameter bounds on graphs (arXiv:1608.07778)
Shiping Liu, Florentin Münch, Norbert Peyerimhoff, 8pp.
• Uniqueness of form extensions and domination of semigroups (arXiv:1608.06798)
Daniel Lenz, Marcel Schmidt, Melchior Wirth, 20pp.
• An autocorrelation and discrete spectrum for dynamical systems on metric spaces (arXiv:1608.05636)
Daniel Lenz, 15pp.
• Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds (arXiv:1608.01955)
Michela Egidi, Shiping Liu, Florentin Münch, Norbert Peyerimhoff, 25pp.
• On the spectrum of operator families on discrete groups over minimal dynamical Systems (arXiv:1606.08353, journal)
Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian Seifert,  Math. Z. 287 (2017), no. 3-4, 993-1007.
• An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs (arXiv:1606.07476)
Daniel Lenz, Peter Stollmann, Gunter Stolz, 27pp.
• Model sets with positive entropy in Euclidean cut and project schemes (arXiv:1605.01167)
Tobias Jäger, Daniel Lenz, Christian Oertel, 30pp.
• Uniqueness of form extensions and domination of semigroups, master's thesis (arXiv:1604.05114)
Melchior Wirth, 55pp.
• Spectral notions of aperiodic order (arXiv:1601.06629)
Michael Baake, Daniel Lenz, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 2, 161-190.

## 2015

• Curvature and higher order Buser inequalities for the graph connection Laplacian (arXiv:1512.08134)
Shiping Liu, Florentin Münch, Norbert Peyerimhoff, 46pp.
• Toeplitz flows and model sets (arXiv:1511.08595, journal)
Michael Baake, Tobias Jaeger, Daniel Lenz, Bull. Lond. Math. Soc. 48 (2016), no. 4, 691-698.
• Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive Substitution (arXiv:1510.00545)
Rostislav Grigorchuk, Daniel Lenz, Tatiana Nagnibeda, London Math. Soc. Lecture Note Ser., 436, Cambridge Univ. Press, Cambridge, 2017.
• General Cheeger inequalities for p-Laplacians on graphs (arXiv:1509.06062, journal)
Matthias Keller, Delio Mugnolo, Nonlinear Anal. 147 (2016), 80-95.
• Note on short time behavior of semigroups associated to selfadjoint operators (arXiv:1509.01993, journal)
Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs,  Bull. Lond. Math. Soc. 48 (2016), no. 6, 935-944.
• Continuity of the spectrum of a field of self-adjoint operators (arXiv:1507.04641, journal)
Siegfried Beckus, Jean Bellissard,  Ann. Henri Poincaré 17 (2016), no. 12, 3425-3442.
• Invariant means on Boolean inverse monoids (arXiv:1503.03733, journal)
Ganna Kudryavtseva, Mark V. Lawson, Daniel H. Lenz, Pedro Resende, Semigroup Forum 92 (2016), no. 1, 77-101.
• Remarks on curvature dimension conditions on graphs (arXiv:1501.05839, journal)Florentin Münch, Calc. Var. Partial Differential Equations 56 (2017), no. 1, Art. 11, 8 pp.
Supervision

## Supervision of dissertational projects - Daniel Lenz

• Ian Zimmermann:  Topic concerning non-linear equations.

• Oleksiy Sukaylo:  Topic concering boundaries of graphs.

• Simon Puchert: Topic concerning  harmonic functions.

• Franziska Sieron: Topic in the field of low complexity subshifts.

• Melchior Wirth: Enropic gradient flow structure of quantum Markov semigroups, University of Jena, March 2020.

• Daniel Sell:  Simple Toeplitz subshifts: combinatorial properties and  uniformity of cocycles,  University of Jena, March 2020.

• Florentin Münch: Dircrete Ricci curvature, diameter bounds´and rigigity, University Potsdam, December 2019, joint supervision with Matthias Keller, Potsdam.

• Marcel Schmidt: Energy forms, Friedrich-Schiller-Universität Jena, May 2017.

• Siegfried Beckus:  Spectral approximation of periodic Schr\"odinger Operators, Friedrich-Schiller-Universität Jena, October 2016

• Felix Pogorzelski: Banach-space valued ergodic theorems and spectral approximation, Friedrich-Schiller-Universität Jena,  October 2014.

• Sebastian Haeseler: Analysis of Dirichlet forms on graphs, Friedrich-Schiller-Universität Jena, November 2013.

• Daniel Wingert, "Evolutionsgleichungen und obere Abschätzungen an die Lösungen des Anfangswertproblems",  TU Chemnitz, Juli 2012; joint supervision with Prof. Dr. Peter Stollmann, TU Chemnitz.

• Carsten Schubert: "Quantengraphen mit zufälligem Potential', TU Chemnitz, December 2011; joint supervision with Prof. Dr. Peter Stollmann, TU Chemnitz.
• Matthias Keller: "On the spectral theory of operators on trees", Friedrich-Schiller Universität Jena, December  2010.
• Adnene Besbes: "Ergodic theorems on aperiodic tilings and applications",  Universite Paris 7, July 2008; joint supervision with Prof. Dr. A. Boutet de Monvel, Universite Paris 7.

• Steffen Klassert: "Random operators on Delone sets", TU Chemnitz,  May 2007;  joint supervision with Prof. Dr. P. Stollmann, TU Chemnitz.

## Supervision of Master theses:

• Ian Zimmermann: On blow-up of solutions for an abstract semilinear parabolic equation, August 2019.

• Oleksiy Sukaylo: Boundaries of hyperbolic graphs, May 2019.

• Simon Puchert: Resolvable graphs and harmonic functions, Juni 2019.

• Sebastian Uschmann: Kohomologie von Dirichletformen, December 2018.

• Fransiska Sieron: The density of periodic configurations in strongly irreducible subshifts of finite type,  February 2017.

• Michael Schwarz: Boundary representations of Dirichlet forms on discrete spaces, November 2015.

• Jannis Koberstein: Surjectivity of graph Laplacians, November 2015.

• Melchior Wirth:  Uniqueness of form extensions and domination of semigroups, October 2015.

• Daniel Sell: Topological groupoids and Matuis spatial realization theorem, September 2015.

• Florentin Münch: Li-Yau inequalities on finite graphs, November 2014, (joint supervision with Dr. M. Keller).

## Supervision of Diploma theses:

• Siegfried Beckus,  "Generalized Bloch Theory for Quasicrystals", Diplom Maerz 2012.

• Marcel Schmidt, "Global properties of Dirichlet forms on discrete spaces", Diplom October 2011.

• Felix Pogorzelski, "Ergodic Theorems on amenable groups", September 2010,  (joint supervision with Prof. Dr. R. Nagel).

• Carsten Schubert, "Laplace-Operatoren auf Quantengraphen", Diplom December 2006, (joint supervision with Prof. Dr. P. Stollmann).

• Matthias Keller, "Produkte zuf\"alliger Matrizen und der Lyapunov-Exponent", Diplom June 2006, (joint supervision with Prof. Dr. P. Stollmann).