The Home Page of Maciej Zworski

Maciej Zworski

Professor of Mathematics

Mathematics Department

University of California, Berkeley

Office: 801 Evans Hall

Fax: (510) 642-8204

Graduate Students

Postdoctoral Researchers

In Fall 2024 I am teaching Math 126, Introduction to PDE and Math 206, Functional Analysis.

In the Fall of 2004 and in the Spring of 2011 I enjoyed two sabbatical semesters and in 2017/2018 a sabbatical year. In the Spring of 2023 I enjoyed another sabbatical semester.

Mathematical theory of scattering resonances, with Semyon Dyatlov.

The schedule of Bay Area Microlocal Analysis Seminar (BAMAS) can be found here.

The schedule of The Paris/Berkeley/Bonn/Zurich Analysis Seminar can be found here.

Information about journals with which I am affiliated can be found below.

Pure and Applied Analysis

Journal of Spectral Theory

American Journal of Mathematics

Methods and Applications of Analysis

I study mathematical aspects of quantum mechanics, more specifically scattering theory and microlocal analysis. CV.

Some recent talks:

Fractal Weyl laws in chaotic scattering , RES04, January 2004.

Control theory and high energy eigenfunctions, (with Nicolas Burq), Forges-les-eaux, June 2004. Here is the short note for the proceedings.

Fractal Weyl laws for open quantum maps , SCATT05, March 2005.

Quantum decay rates in chaotic scattering , Talk at Ecole Polytechnique, May 2006.

Soliton scattering by delta impurities , Colloque EDP Evian-les-Bains, June 2006.

What are the residues of resolvents of Laplacians on symmetric spaces? , Summers School: Microlocal and Geometric Methods in Representation Theory, July 2006.

Quantum decay rates in chaotic scattering (with Stéphane Nonnenmacher), AMS National Meeting, New Orleans, January 2007.

Quantum chaos in scattering theory Colloquium talks 2006/08 at First Joint CMS/SMM Meeting , Northwestern, Univ. of North Carolina, and Univ. of Washington.

Soliton Home Movies, a popular talk for prospective graduate students (a reasonably decent PDF viewer might be needed to see the movies)

A minicourse at The Summer Graduate Worskhop on Inverse Problems, July 2009: I, II, III, IV, V.

Two solitons in an external field, Colloque Franco-Tunisien d'EDP, Hammamet, 2009.

Mathematics of quantum resonances, The Joint BMC/BAMC Meeting of 2010.

Semiclassical scattering and applications, Colloquium, Stanford, January 2011.

Solitons in external fields, Séminaire EDP, Ecole Polytechnique/IHES, February 2011.

From classical to quantum and back, Colloquium at Banach Center Warsaw and at Purdue University.

Linear vs chaotic, talk at 2016 Breakthrough Prize Symposium: A Multilingual Conversation in Science.

Internal waves in fluids and spectral theory of 0th order operators, talks at Villa Finaly and at University of Warsaw.

Theory and computation of resonances in 1d scattering: David Bindel's website (still in progress and developing) with his easy-to-use MATLAB codes for computations of resonances.

My 1999-2008 papers can be found here.

My more recent papers can be found below.

Overdamped QNM for Schwarzschild black holes (with Michael Hitrik), June 2024.
Discrete vs. continuous in the semiclassical limit, (with Simon Becker and Jens Wittsten), June 2024.

Classical--Quantum correspondence in Lindblad evolution (with Jeff Galkowski and with an appendix joint with Zhen Huang), June 2024.
Optimal enhanced dissipation for contact Anosov flows (with Zhongkai Tao), October 2023.
Classically forbidden regions in the chiral model of twisted bilayer graphene (with Michael Hitrik and with an appendix joint with Zhongkai Tao), October 2023.
Mathematical results on the chiral model of twisted bilayer graphene (with an appendix by Mengxuan Yang and Zhongkai Tao), October 2023, to appear in J. Spectr. Theory.
From the chiral model of TBG to the Bistritzer--MacDonald model, (with Simon Becker), J. Math. Phys., 65 (2024), no. 6, Paper No. 062103, 13 pp.

An abstract formulation of the flat band condition (with Jeff Galkowski), July 2023, to appear in AMS Contemporary Mathematics volume in honor of Steve Zelditch.

Degenerate flat bands in twisted bilayer graphene, (with Simon Becker and Tristan Humbert), June 2023.

Dirac points for twisted bilayer graphene with in-plane magnetic field, (with Simon Becker), J. Spectr. Theory, 14 (2024), no. 2, 479–511.

The scattering phase: seen at last, (with Jeff Galkowski, Pierre Marchand and Jian Wang), October 2022, SIAM Journal on Applied Matematics, 84(2024), 246–261.

Integrability in the chiral model of magic angles, (with Simon Becker and Tristan Humbert), Comm. Math. Phys. 403(2023), 1153--1169.

Fine structure of flat bands in a chiral model of magic angles, (with Simon Becker and Tristan Humbert), August 2022.

Mathematics of internal waves in a 2D aquarium, (with Semyon Dyatlov and Jian Wang), December 2021, to appear in Analysis & PDE.

Viscosity limits for 0th order pseudodifferential operators, (with Jeff Galkowski), Communications on Pure and Applied Mathematics, 75(2022), 1798-1869.

An introduction to microlocal complex deformations, (with Jeff Galkowski), an expository companion to the paper above; December 2019.

Spectral characterization of magic angles in twisted bilayer graphene, (with Simon Becker, Mark Embree and Jens Wittsten), Physics Review B, 103, 165113, 2021.

Mathematics of magic angles in a model of twisted bilayer graphene, (with Simon Becker, Mark Embree and Jens Wittsten), Probability and Mathematical Physics, 3 (2022), no. 1, 69–103.

Honeycomb structures in magnetic fields, (with Simon Becker, Rui Han and Svetlana Jitomirskaya), Journal of Physics A, 2021, vol.54, no.34, 345203.

Outgoing solutions via Gevrey-2 properties, (with Jeff Galkowski), Annals of PDE (2021), no. 1, Paper No. 5, 13 pp.

Analytic hypoellipticity of Keldysh operators, (with Jeff Galkowski), Proceedings of the London Mathematical Society, (3)(2021), 1--19.

Fractal uncertainty for transfer operators (with Semyon Dyatlov), International Mathematics Research Notices, 2020, no. 3, 781--812.

Microlocal analysis of forced waves, (with Semyon Dyatlov), Pure and Applied Analysis, 1(2019), 359--394.

Magnetic oscillations in a model of graphene, (with Simon Becker), Comm. Math. Phys. 367(2019), 941--989.
An addendum to this paper clarifying an issue around Bohr--Sommerfeld quantization conditions, July 2019.

Rough controls for Schrödinger operators on tori, (with Nicolas Burq), Annales Henri Lebesgue, 2 (2019), 331--347.

Remarks on Vasy's operator with analytic coefficients (with Gilles Lebeau), Proceedings of the AMS, 147(2019), 145--152.

Commentary on ``Differentiable dynamical systems'' by Stephen Smale, Bulletin of the AMS, 55 (2018), 331--336.

A semiclassical approach to the Kramers--Smoluchowski equation (with Laurent Michel), SIAM Journal of Mathematical Analysis, 50(2018), 5362--5379.

Resonances for obstacles in hyperbolic space (with Peter Hintz), Comm. Math. Phys. 359(2018), 699--731.

Scattering resonances as viscosity limits, in "Algebraic and Analytic Microlocal Analysis", M. Hitrik, D. Tamarkin, B. Tsygan, and S. Zelditch, eds. Springer, 2018.
A numerical illustration of the results and a MATLAB code can be found here.
Wave decay for star-shaped obstacles in R3: papers of Morawetz and Ralston revisited, (with Peter Hintz), Mathematical Proceedings of the Royal Irish Academy, 117A (2017), 47--62.

Mathematical study of scattering resonances, Bulletin of Mathematical Sciences, 7(2017), 1--85.

Ruelle zeta function at zero for surfaces (with Semyon Dyatlov), Inventiones mathematicae, 210(2017), 211--229.

A local trace formula for Anosov flows (with Long Jin) with an appendix by Frédéric Naud, Ann. Inst. Henri Poincaré (A), 18(2017), 1-35.
I am grateful to Long Jin and Zhongkai Tao for filling a gap, and correcting a mistake, in the proof in a recent paper.
A Fermi golden rule for quantum graphs (with Minjae Lee), Journal of Mathematical Physics, 57, 092101 (2016).
Resonances for asymptotically hyperbolic manifolds: Vasy's method revisited, Journal of Spectral Theory, 6(2016), 1087--1114.
A less formal account can be found here:
Dynamical zeta function for Anosov flows via microlocal analysis (with Semyon Dyatlov), Annales de l'ENS, 49(2016), 543--577.

Heat traces and existence of scattering resonances for bounded potentials (with Hart F. Smith), Annales de l'Institut Fourier, 66 (2016), 455--475.

Stochastic stability of Pollicott-Ruelle resonances (with Semyon Dyatlov), Nonlinearity, 28(2015), 3511--3534.

Resonances and lower resolvent bounds (with Kiril Datchev and Semyon Dyatlov), Journal of Spectral Theory, 5(2015), 599--615.

Decay of correlations for normally hyperbolic trapping, (with Stéphane Nonnenmacher), Inventiones mathematicae, 200(2)(2015), 345--438.

Fractal Weyl law for open quantum chaotic maps (with Stéphane Nonnenmacher and Johannes Sjöstrand), Annals of Math. 179(1)(2014), 179--251.

Sharp polynomial bounds on the number of Pollicott--Ruelle resonances (with Kiril Datchev and Semyon Dyatlov), Ergodic Theory and Dynamical Systems, 34(4)(2014), 1168--1183.

Trapping of waves and null geodesics for rotating black holes (with Semyon Dyatlov), Phys. Rev. D 88, 084037 (2013)

Control for Schrödinger operators on 2-tori: rough potentials, (with Jean Bourgain, and Nicolas Burq), Journal of the European Mathematical Society, 15(2013), no.5, 1597--1628. Erratum.

Experimental observation of spectral gap in microwave n-disk systems (with Sonya Barkhofen, Tobias Weich, Alexander Potzuweit, Ulrich Kuhl and H.-J. Stöckmann), Phys. Rev. Lett. 110, 164102 (2013).

Pointwise bounds on quasimodes of semiclassical Schrödinger operators in dimension two (with Hart F. Smith), Math. Res. Lett. 20(2013), 401--408.

Weyl asymptotics: from closed to open systems (with Alexander Potzuweit, Tobias Weich, Sonya Barkhofen, Ulrich Kuhl and H.-J. Stöckmann), Phys. Rev. E. 86, 066205 (2012).

A quantitative version of Catlin-D'Angelo-Quillen theorem (with Alexis Drouot), Analysis and Mathematical Physics, 3(2013), no.1, 1-19.

Quantum ergodicity for restrictions to hypersurfaces (with Semyon Dyatlov), Nonlinearity, 26(2013), 35-52.

Weighted eigenfunction estimates with applications to compressed sensing (with Nicolas Burq, Semyon Dyatlov and Rachel Ward), SIAM J. Math. Anal., 44(5)(2012), 3481--3501.

Control for Schrödinger operators on tori, (with Nicolas Burq), Math. Res. Lett. 19(2)(2012), 309-324.

From open quantum systems to open quantum maps (with Stéphane Nonnenmacher and Johannes Sjöstrand), Comm. Math. Phys. 304(1)(2011), 1-48.

Resolvent estimates for normally hyperbolic trapped sets (with Jared Wunsch), Ann. Inst. Henri Poincaré (A), 12(7)(2011), 1349-1385.

Effective dynamics of double solitons for perturbed mKdV(with Justin Holmer and Galina Perelman), with an appendix by Bernd Sturmfels, Comm. Math. Phys. 305(3)(2011), 363-425.

Numerical illustrations of the results and MATLAB codes can be found here. Related numerical results can be found in a paper by Trevor Potter.

Probabilistic Weyl laws for quantized tori(with Tanya J. Christiansen), Comm. Math. Phys. 299(2)(2010), 305-334.

Semiclassical resolvent estimates in chaotic scattering (with Stéphane Nonnenmacher ), Applied Mathematics Research eXpress 2009; doi: 10.1093/amrx/abp003

A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix(with Tanya J. Christiansen), J. Phys. A: Math. Theor., 42 (2009) 415202.

Breathing patterns in nonlinear relaxation(with Justin Holmer), Nonlinearity, 22(2009), 1259-1301.

Quantum decay rates in chaotic scattering (with Stéphane Nonnenmacher ), Acta Mathematica, Volume 203, Number 2/December, 2009, 149-233.