Recent Progress in Path Integration on Graphs and Manifolds (organized together with Matthias Keller, Kazumasa Kuwada, Anton Thalmaier) Oberwolfach Mini-Workshop, 7 – 13 April 2019
Covariant Schrödinger semigroups on noncompact Riemannian manifolds. Published as a monograph in the Birkhäuser series 'Operator theory: advances and applications'. The book can be obtained here.
Book chapter:
Geometric Analysis on Singular Spaces (w. Francesco Bei, Jochen Brüning, Matthias Ludewig; 68 pages). Appeared as a chapter in the book 'Space - Time - Matter: Analytic and Geometric Structures (SFB 647)', De Gruyter Verlag.
The book can be obtained here.
Editorial activities:
Einstein Equations: Physical and Mathematical Aspects of General Relativity. Sergio Cacciatori, Batu Güneysu, Stefano Pigola (Eds.). Published as proceedings to the summerschool 'Domoschool 2018' in the Birkhäuser series 'Tutorials, Schools, and Workshops in the Mathematical Sciences'. The book can be obtained here.
Feynman-Kac formula for perturbations of order $\leq 1$, and noncommutative geometry (with Sebastian Boldt).
Preprint version
Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators (with Sebastian Boldt).
Preprint version
Heat flow regularity, Bismut--Elworthy--Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature (with Mathias Braun).
Preprint version
The Chern character of theta-summable Fredholm modules over dg algebras and localization on loop space (with Matthias Ludewig).
Preprint version
Heat kernel estimates and the relative compactness of perturbations by potentials (with Jochen Brüning).
Preprint version
Papers published or accepted in peer-reviewed journals :
Hölder estimates for magnetic Schrödinger semigroups from mirror coupling (with Oliver Fürst).
Preprint version To appear in Letters in Mathematical Physics.
RCD*(K,N) spaces and the geometry of multi-particle Schrödinger semigroups.
Preprint version To appear in International Mathematics Research Notices.
Molecules as metric measure spaces with Kato-bounded Ricci curvature (with Max von Renesse). Preprint version To appear in Comptes Rendus Mathématique.
On the geometry of semiclassical limits on Dirichlet spaces.
Preprint version To appear in Journal of Functional Analysis.
Kac regular sets and Sobolev spaces in geometry, probability and quantum mechanics (with Francesco Bei). To appear in Mathematische Annalen.
Preprint version
Odd characteristic classes in entire cyclic homology
and equivariant loop space homology (with Sergio Cacciatori). To appear in Journal of Noncommutative Geometry.
Preprint version
Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow (with Anton Thalmaier). Ann. Inst. Fourier (Grenoble) 70 (2020), no. 1, 437–456. 35 (58)
Link to paper on journal homepage.
On the semimartingale property of Brownian bridges on complete manifolds. ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019), no. 1, 15–31.
Link to paper on journal homepage.
Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs (with Matthias Keller). Journal d'Analyse Mathematique 141 (2020), no. 2, 751–770.
Link to paper on journal homepage
Scattering the geometry of weighted graphs (with Matthias Keller). Mathematical Physics, Analysis and Geometry volume 21, Article number: 28 (2018).
Link to paper on journal homepage.
L^p interpolation inequalities and global Sobolev regularity results (with Stefano Pigola and an appendix by Ognjen Milatovic). Annali di Matematica Pura ed Applicata (1923 -) Volume 198 (2019), Issue 1, pp 83–96.
Link to paper on journal homepage.
Nonlinear Calderon-Zygmund inequalities for maps (with Stefano Pigola). Ann. Global Anal. Geom. 54 (2018), Issue 3, pp 353–364.
Link to paper on journal homepage.
Quantitative C^1 estimates on manifolds (with Stefano Pigola). To appear in International Mathematics Research Notices 2018, no. 13, 4103–4119. Link to paper on journal homepage.
Heat kernels in the context of Kato potentials on arbitrary manifolds. Potential Anal. 46 (2017), no. 1, 119--134.
Link to paper on journal homepage.
q-Parabolicity of stratified pseudomanifolds and
other singular spaces (with Francesco Bei). Ann. Global Anal. Geom. 51 (2017), no. 3, 267--286.
Link to paper on journal homepage.
Scattering theory of the Hodge Laplacian under a conformal perturbation (with Francesco Bei and Joern Mueller). J. Spectr. Theory 7 (2017), no. 1, 235--267.
Link to paper on journal homepage.
On Some Generalisations of Meyers-Serrin Theorem} (with Davide Guidetti and Diego Pallara). Bruno Pini Mathematical Analysis Seminar (2015). doi: 10.6092/issn.2240-2829/5894.
Link to paper on journal homepage.
The Calder\'on-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds (with Stefano Pigola). Adv. Math. 281 (2015), 353–393.
Link to paper on journal homepage.
L^1-elliptic regularity and H=W on the whole
L^p-scale on arbitrary manifolds (with Davide Guidetti and Diego Pallara). Ann. Acad. Sci. Fenn. Math. 42 (2017), no. 1, 497–521.
Link to paper on journal homepage.
Stochastic differential equations for models of nonrelativistic matter interacting with quantized radiation fields (with Oliver Matte and Jacob Schach-Møller). To appear in Probability Theory and Related Fields, pp. 1--99, DOI 10.1007/s00440-016-0694-4.
Link to paper on journal homepage.
Semiclassical limits of quantum partition functions on infinite graphs. J. Math. Phys. 56 (2015), no. 2, 022102, 13 pp.
Link to paper on journal homepage.
The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE's (with Markus Pflaum). SIGMA 13 (2017), 003, 44 pages
Link to paper on journal homepage.
Generalized Schrödinger semigroups on infinite graphs (with Ognjen Milatovic and Francoise Truc). Potential Anal. 41 (2014), no. 2, 517–541.
Link to paper on journal homepage.
A Feynman-Kac-Ito Formula for magnetic Schrödinger operators on graphs (with Matthias Keller and Marcel Schmidt).
To appear in Probability Theory and Related Fields, pp. 1--35, DOI 10.1007/s00440-015-0633-9. Link to paper on journal homepage.
Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below (with Diego Pallara).
Math. Ann. 363 (2015), no. 3-4, 1307–1331. Link to paper on journal homepage.
Path integrals and the essential self-adjointness of differential operators on noncompact manifolds (with Olaf Post). Math.Z. 275 (2013), Issue 1-2, pp 331--348. Link to paper on journal homepage.
Nonrelativistic Hydrogen type stability problems on nonparabolic 3-manifolds.
Annales Henri Poincaré 13 (2012), 1557--1573. Link to paper on journal homepage.
Kato's inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds.
To appear in Proc. of the Amer. Math. Soc (2012). (Final) preprint version
Multiplicative matrix-valued functionals and continuity properties
of semigroups corresponding to differential operators with matrix
coefficients.
J. Math. Anal. Appl. 380 (2011), 709--725. Link to paper on journal homepage.
The Feynman-Kac formula for Schrödinger operators on vector bundles over complete manifolds.
J. Geom. Phys 60 (2010), 1997--2010.
The representation of strongly continuous operator semigroups on vector bundles as Hida distributions.
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 3, 511--519.
Papers published in peer-reviewed conference proceedings:
Stochastic calculus and nonrelativistic quantum electrodynamics (with Oliver Matte and Jacob Schach-Møller). To appear in the Conference Proceedings of QMATH 12
Preprint version
Recent probabilistic results on covariant Schrödinger operators on infinite weighted graphs (with Ognjen Milatovic). To appear in the Conference Proceedings of QMATH 12
Preprint version
Theses:
Covariant Schrödinger semigroups on noncompact Riemannian manifolds (Habilitationsschrift).
Humboldt-Universität zu Berlin. An extended version of this thesis has appeared as a book in Birkhäuser.
On the Feynman-Kac formula for Schrödinger semigroups on vector bundles (Dissertation).
Bonn. This is a slightly revised version of my PhD thesis (under the supervision of Prof. Dr. Matthias Lesch). It includes an introduction to stochastic analysis (on manifolds).
File
Ein Kriterium für die Absolutstetigkeit des Spektrums von Sturm-Liouville-Operatoren (Diplomarbeit).
Frankfurt am Main. Under the supervision of Prof. Dr. Joachim Weidmann.