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Semiclassical measures for complex hyperbolic quotients,
with Jayadev Athreya and Nicholas Miller
We study semiclassical measures on compact quotients of the complex hyperbolic space.
We show that the support of each such measure is either the entire cosphere bundle
or it contains the cosphere bundle of a compact immersed totally geodesic complex submanifold.
We use the strategy of the previous paper
and a classification of orbit closures of fast unstable/stable vector fields.
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Semiclassical measures for higher dimensional quantum cat maps,
with Malo Jézéquel,
Annales Henri Poincaré 25(2024), 1545–1605
We investigate semiclassical measures for higher dimensional quantum cat maps, which
are standard toy models in quantum chaos. If the matrix quantized has a spectral gap (i.e. a simple
eigenvalue of largest absolute value) and its characteristic polynomial is irreducible over
the rationals, then we show that all semiclassical measures have full support. The spectral gap assumption
makes it possible for us to still use the one-dimensional fractal uncertainty principle in this higher
dimensional situation. When the characteristic polynomial is reducible over rationals, we obtain
a more complicated to state result and compare it with existing counterexamples.
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Control of eigenfunctions
on surfaces of variable curvature,
with Long Jin
and Stéphane Nonnenmacher,
Journal of the American Mathematical Society 35(2022), 361–465
We show lower bounds on mass of eigenfunctions on negatively curved surfaces,
and more generally surfaces with Anosov geodesic flow. We also obtain observability/control for the Schrödinger equation and exponential decay for the damped wave equation.
The proofs partially use the strategy of the previous work in constant curvature, however the case of variable curvature presents substantial challenges due to which a large part of the proof proceeds in a very different way. [Diagram of dependence of sections]
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Semiclassical measures on hyperbolic surfaces have full support,
with Long Jin,
Acta Mathematica 220(2018), 297–339
Using the fractal uncertainty principle proved in a previous paper with Bourgain,
we show that every semiclassical defect measure on a compact hyperbolic surface
has support equal to the entire cosphere bundle.
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Fractal uncertainty for transfer operators,
with Maciej Zworski,
International Mathematics Research Notices (2020), 781–812.
We show that fractal uncertainty principle implies an essential spectral gap
for convex co-compact hyperbolic surfaces. This partially recovers
the result of the paper with Zahl.
In contrast to the latter paper, which relied on complicated microlocal techniques,
our proof uses transfer operators and is relatively short and self-contained.
- Fourier dimension and spectral gaps for hyperbolic surfaces,
with Jean Bourgain, Geometric and Functional Analysis 27(2017), 744–771
For a convex co-compact hyperbolic surface with limit set Λ of dimension δ>0, we show that the Fourier dimension of Λ is at least ε for some ε>0 depending only on δ. The proof uses the discretized sum-product theorem in R and the nonlinear nature of the transformations in the group Γ; in fact,
our result would be false for linear Cantor sets. Using fractal uncertainty principle,
we deduce an essential spectral gap of size 1/2-δ+ε.
- Dolgopyat's method and the fractal uncertainty principle,
with Long Jin,
Analysis & PDE 11(2018), 1457–1485
We show that every Ahlfors–David regular subset of R of positive dimension
satisfies the fractal uncertainty principle with exponent strictly larger than the one coming from the volume bound.
The method of proof is inspired by the works of Dolgopyat, Naud, and Stoyanov in the context
of spectral gaps for transfer operators, and can be crudely summarized as follows:
triangle inequality in a Hilbert space is rarely sharp. As an application we obtain
new spectral gaps for convex co-compact hyperbolic surfaces and open quantum baker's maps.
- Spectral gaps without the pressure condition,
with Jean Bourgain,
Annals of Mathematics 187(2018), 825–867
We show that every convex co-compact hyperbolic surface has an essential spectral gap.
The proof uses the approach developed in the paper with Zahl
and the key new component is the fractal uncertainty principle for δ-regular sets with all δ<1. Previously such gaps were known only under the relaxed pressure
condition δ≤1/2.
- Resonances for open quantum maps and a fractal uncertainty principle,
with Long Jin,
Communications in Mathematical Physics 354(2017), 269–316
We study open quantum baker's maps, which are models in open quantum chaos having Cantor sets as their trapped sets.
Using the fractal uncertainty approach developed in the paper with Zahl
and the arithmetic structure of Cantor sets, we show that all such systems have a spectral gap,
and obtain quantitative bounds on the size of the gap. We also show
an improved Weyl bound similar to the one here.
Both results are supported by numerical evidence.
- Improved fractal Weyl bounds for hyperbolic manifolds,
with an appendix with David Borthwick and Tobias Weich,
Journal of the European Mathematical Society 21(2019), 1595–1639
We give a Weyl upper bound on the number of scattering resonances on convex co-compact hyperbolic quotients in strips. The exponent in the bound depends on the width of the strip and improves on the standard Weyl upper bound, in particular this exponent is negative until the Patterson–Sullivan gap. The appendix gives numerical evidence for the new Weyl bound.
- Spectral gaps, additive energy, and a fractal uncertainty principle,
with Joshua Zahl,
Geometric and Functional Analysis 26(2016), 1011–1094
We obtain an improved spectral gap for convex co-compact hyperbolic surfaces with the dimension of the limit set close to 1/2.
Compared to previous works on spectral gaps which rely on Dolgopyat's method, we decouple the combinatorial difficulties of
the problem from the analytical ones and obtain an explicit formula for the gap in terms of additive energy of the limit set,
which in turn is estimated using its Ahlfors–David regularity.
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Mathematics of internal waves in a 2D aquarium,
with Jian Wang and
Maciej Zworski,
Analysis & PDE
18(2025), 1–92
We work with a linearized model of fluid in an effectively two-dimensional aquarium. Under the Morse–Smale assumption
on the chess billiard map, we show a limiting absorption principle and apply it to get long time asymptotics of the solution
to the evolution equation, showing that the singular part concentrates on the attractor of the chess billiard flow similarly to
this experimental video from ENS Lyon.
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Microlocal analysis of forced waves,
with Maciej Zworski,
Pure and Applied Analysis 1(2019), 359–384
We study a forcing problem for a 0th order pseudodifferential operator on a surface whose reduced Hamiltonian flow is a Morse–Smale flow without fixed points. In particular we explain how some of the results of a recent paper of Colin de Verdière and Saint-Raymond can be obtained using
radial estimates.
- Fractal Weyl laws and wave decay for general trapping,
with Jeffrey Galkowski,
Nonlinearity
30(2017), 4301–4343
We give an upper bound on the number of resonances in strips for general
manifolds with Euclidean infinite ends, without any assumptions on the trapped set.
Our bound depends on the volume of the set of trajectories which are trapped for
the Ehrenfest time, and gives a polynomial improvement over the standard bound
Rd-1 when the escape rate is positive. We also prove a wave decay result with
high probability for random initial data.
- Lower resolvent bounds and Lyapunov exponents,
with Alden Waters, Applied Mathematics Research Express 2016, 68–97
We obtain a polynomial lower bound on the norm of the scattering resolvent in the lower half-plane under very mild assumptions on trapping, namely existence of a trajectory which is trapped in the past, but not in the future.
The power in the bound, depending on the maximal expansion rate along the trajectory,
gives the smallest number of derivatives lost in an exponential decay of local energy estimate
for the wave equation.
- Resonances and lower resolvent bounds,
with Kiril Datchev and Maciej Zworski,
Journal of Spectral Theory 5(2015), 599–615
We show that in the presence of certain elliptic trapping, one-sided bounds on the resolvent with a cutoff supported far away into the infinity are at best exponential. This
is in contrast with the known polynomial two-sided bounds.
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Resonance projectors and asymptotics
for r-normally hyperbolic trapped sets, Journal of the American Mathematical Society
28(2015), 311–381.
We show existence of a band of resonances with a Weyl law when the trapped
set is r-normally hyperbolic for large r and the normal expansion
rates are half-pinched. This dynamical setting is stable under perturbations and is applied
the next paper
to Kerr–de Sitter black holes. The key tool is a Fourier integral operator microlocally projecting
onto resonant states in the band. This operator lets us microlocally represent resonance expansions as
Taylor expansions; a variety of methods from microlocal analysis, especially positive commutator estimates, finish the proof.
The r-normal hyperbolicity assumption is explicitly needed in the construction of the projector to ensure smoothness
of its symbol.
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Fractal Weyl laws for asymptotically hyperbolic manifolds,
with Kiril Datchev,
Geometric and Functional Analysis 23(2013), 1145–1206.
We obtain a fractal upper bound on the number of resonances in disks of fixed size centered at
the unitarity axis for a general class of manifolds, including convex co-compact hyperbolic quotients.
The exponent in the bound is related to the dimension of the trapped set.
We use the techniques of Sjöstrand–Zworski
on resonance counting for hyperbolic trapped sets and of
Vasy
on effective meromorphic continuation of the resolvent for asymptotically hyperbolic
infinite ends.
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Symmetry of bound and antibound states in the semiclassical limit
for a general class of potentials, with
Subhroshekhar Ghosh,
Proceedings of the American Mathematical Society 138 (2010), 3203–3210.
It was proved by Bindel and Zworski in this paper
that for scattering on the half-line by a compactly supported
potential with a constant bump at the end of its support,
the purely imaginary poles of the resolvent
become symmetric with respect to the real axis modulo
errors exponentially small in the semiclassical limit.
Our paper is an extension
of this result to a more general class of potentials,
allowing any positive bump, and also provides a different
explanation of why this phenomenon holds.
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Pollicott–Ruelle resolvent and Sobolev regularity,
Pure and Applied Functional Analysis 8(2023), 187–213.
We use radial estimates (a version for vector bundles)
to investigate what Sobolev regularity is needed for the meromorphic continuation of the Pollicott–Ruelle resolvent for an Anosov flow, in case of an arbitrary
lift of the flow to a vector bundle. The resulting condition is similar
to the one obtained in the paper of
Bonthonneau–Lefeuvre for Hölder–Zygmund regularity.
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Ruelle zeta function at zero for nearly hyperbolic 3-manifolds,
with Mihajlo Cekić, Benjamin Delarue, and Gabriel Paternain,
Inventiones Mathematicae 229(2022), 303–394
We study the vanishing order of the Ruelle zeta function at zero for a compact connected oriented
negatively curved 3-manifold. We show that if the metric is a small perturbation of the hyperbolic metric,
then the order of vanishing is equal to 4-b1 where b1 is the first Betti number.
This is in contrast with the hyperbolic case where this order is 4-2b1 and with
the case of surfaces where the previous paper with Zworski showed that the order only depends
on the genus, not on the metric.
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Dynamical zeta functions for Axiom A flows,
with Colin Guillarmou,
Bulletin of the American Mathematical Society 55(2018), 337–342
We show that the meromorphic continuation result for Ruelle zeta function
of our previous paper
applies to Axiom A flows with orientable stable/unstable spaces,
using results of Conley–Easton and
Guillarmou–Mazzucchelli–Tzou.
- Ruelle zeta function at zero for surfaces,
with Maciej Zworski,
Inventiones Mathematicae 210(2017), 211–229
Using previous work with Zworski as well
as relatively standard microlocal techniques,
we show that the Ruelle zeta function for
a negatively curved oriented surface vanishes at zero to the order
given by the absolute value of the Euler characteristic.
This result, giving a relation between dynamics
and topology, was previously known only in constant curvature via the Selberg trace formula.
- Pollicott–Ruelle
resonances for open systems,
with Colin Guillarmou,
Annales Henri Poincaré 17(2016), 3089–3146
We define Pollicott–Ruelle resonances for hyperbolic flows on noncompact manifolds (in a setting
closely related to basic sets of Axiom A flows). They arise either as poles of Fourier–Laplace transforms
of correlations or as poles of meromorphic extensions of dynamical zeta functions. Our construction is based
on the microlocal approach of Faure–Sjöstrand
and Dyatlov–Zworski.
- Stochastic stability of Pollicott–Ruelle resonances,
with Maciej Zworski,
Nonlinearity 28(2015), 3511–3534
We show that Pollicott–Ruelle resonances of an Anosov flow are the limits of the L2 eigenvalues
of the generator of the flow with an additional damping term. The latter eigenvalues characterize decay of correlations
for a stochastic modification of the flow.
- Power spectrum of the geodesic flow on hyperbolic manifolds,
with Frédéric Faure and Colin Guillarmou,
Analysis & PDE 8(2015), 923–1000.
We describe the Pollicott–Ruelle resonances for compact hyperbolic quotients in all dimensions. In dimensions greater than 2,
this description is much more involved than for surfaces and features in particular the spectrum of the Laplacian on trace-free
divergence-free symmetric tensors of all orders.
- Dynamical zeta functions for Anosov flows via microlocal analysis,
with Maciej Zworski,
Annales de l'ENS 49(2016), 543–577
We prove meromorphic continuation of the Ruelle dynamical zeta function for Anosov flows, recovering
recent results of Giulietti, Liverani, and Pollicott.
We follow the approach of Faure–Sjöstrand; the key new component is the restriction on the wavefront
set of the resolvent, using the methods of Melrose and Vasy, making it possible to take the flat trace.
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Sharp polynomial bounds on the number of Pollicott–Ruelle resonances,
with Kiril Datchev
and Maciej Zworski,
Ergodic Theory and Dynamical Systems 34(2014), 1168–1183.
Using the methods of the previous paper,
we obtain sharp upper bounds on the number of Ruelle resonances,
defined using the framework of Faure
and Sjöstrand.
- Spectral gaps for normally hyperbolic trapping,
Annales de l'Institut Fourier 66(2016), 55–82
Motivated by Kerr and Kerr–de Sitter black holes, we show existence of spectral gaps of optimal size for normally hyperbolic
trapped sets with codimension 1 smooth stable/unstable foliations. The short proofs, based on semiclassical defect measures,
simplify and refine some of the earlier results of Wunsch–Zworski,
Nonnenmacher–Zworski (in our special case), and
myself.
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Trapping of waves and null geodesics for rotating black holes,
with Maciej Zworski,
Physical Review D
88(2013), 084037.
A note outlining some of the recent work on resonances for rotating black holes.
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Asymptotics of linear waves and resonances
with applications to black holes, Communications in Mathematical Physics 335(2015), 1445–1485.
We apply the methods and results of a previous paper
to the setting of Kerr–de Sitter black holes, proving a Weyl law for resonances and an asymptotic
decomposition of linear waves for high frequencies. The latter fact, related to resonance expansion,
is also proved for Kerr black holes using the work of Vasy–Zworski.
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Asymptotic distribution of quasi-normal modes for
Kerr–de Sitter black holes,
Annales Henri Poincaré 13(2012), 1101–1166.
We establish a Bohr–Sommerfeld type quantization condition for quasi-normal
modes of a slowly rotating Kerr–de Sitter black hole, observing in particular
a Zeeman-like splitting once spherical symmetry is broken. We compute the resulting
pseudopoles numerically [click here for MATLAB codes and data]
and compare them to those
numerically studied by physicists.
Finally, we prove a resonance decomposition of linear waves.
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Exponential energy decay for Kerr–de Sitter
black holes beyond event horizons,
Mathematical Research Letters
18 (2011), 1023–1035.
The red-shift effect and a parametrix near the event horizons are used
to extend the exponential decay proved in the previous paper
to the whole space.
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Quasi-normal modes and exponential energy decay for the
Kerr–de Sitter black hole,
Communications in Mathematical Physics
306 (2011), 119–163.
This paper constructs an analogue of the scattering resolvent in the case of
a slowly rotating Kerr–de Sitter black hole. The poles are proven to form
a discrete set; they are the
quasi-normal modes
of black holes that have been
numerically studied by physicists.
Using the recent result by Wunsch and Zworski,
we prove existence of a resonance free strip and exponential energy decay for the wave
equation on a fixed compact set.
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Scattering phase asymptotics with fractal remainders,
with Colin Guillarmou,
Communications in Mathematical Physics
324(2013), 425–444.
Using our previous work, we derive scattering
phase asymptotics for manifolds with asymptotically Euclidean ends with remainders that
are fractal for hyperbolic flows. We next derive asymptotics with a similar remainder
for hyperbolic quotients, and non-sharp lower bounds on the remainder in some cases.
Finally, we investigate the relation of scattering phase asymptotics with counting resonances
near the unitarity axis, in particular with the fractal Weyl law.
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Microlocal limits of plane waves and Eisenstein functions,
with Colin Guillarmou,
Annales de l'ENS 47(2014), 371–448.
For Riemannian manifolds that are either Euclidean or hyperbolic near infinity, and
with trapped set of Liouville measure zero, we
show that plane waves, also known as Eisenstein functions, converge to some semiclassical
measure, if one averages in frequency in an h-sized window and in
the direction of the wave. The speed of convergence is estimated in terms
of the classical escape rate and the Ehrenfest time; in many cases,
this is a power of h (in contrast with quantum ergodicity on compact manifolds,
where the best known result is 1/|log h|). As an application,
we derive a local Weyl law for spectral projectors with a fractal remainder.
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Quantum ergodicity for restrictions to hypersurfaces,
with Maciej Zworski,
Nonlinearity 26(2013), 35–52.
We show that on a compact Riemannian manifold with ergodic geodesic flow,
restrictions of the eigenfunctions of the Laplacian to any hypersurface
satisfying a simple geometric condition are equidistributed in phase space.
This work generalizes a paper of
John Toth
and Steve Zelditch using
the methods of semiclassical analysis, and provides a shorter proof.
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Microlocal limits of Eisenstein functions
away from the unitarity axis, Journal of Spectral Theory
2 (2012), 181–202.
We consider a Riemannian surface with cusp ends and show that the Eisenstein functions
in the upper half-plane, away from the real line, converge to a certain canonical
measure. This statement is similar to quantum unique ergodicity(QUE); however,
being away from the real line considerably simplifies the problem. In particular,
no global dynamical properties of the flow are used. As an application,
we prove that the scattering matrix converges to zero in any strip away from the real line.
[ First version, using a more direct analysis
in the cusp ]
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A quick introduction to b-calculus, with Zhenhao Li
For a series of lectures at MSRI. A quick introduction to b-calculus techniques and
an outline of the proof of the Fredholm property of Laplacians on asymptotically cylindrical manifolds on weighted b-Sobolev spaces.
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Lecture notes on distributions, elliptic regularity, and applications to PDEs
For the course 18.155 at MIT. Starts with definition of distributions and ends with Hodge's Theorem.
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Lecture notes on Pollicott–Ruelle resonances and dynamical zeta functions:
Lecture 1, Lecture 2
For the minicourse taught at Geometry and Analysis Seminar for Boston Area Graduate Students
in Fall 2022.
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Lecture notes on semiclassical analysis
For the minicourse taught at the Summer Northwestern Analysis Program in Summer 2019.
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Notes on hyperbolic dynamics
These notes give a self-contained proof of the Stable/Unstable Manifold Theorem (also known
as the Hadamard–Perron Theorem) in hyperbolic dynamics. They also give two examples of hyperbolic systems:
geodesic flows on negatively curved surfaces and dispersing billiards. The proof of the Stable/Unstable Manifold Theorem
starts with a basic model case which however retains the essence of the general case. There are many figures throughout
the text, and it is indended as a (somewhat) gentle introduction to some techniques in hyperbolic dynamics.
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Lecture notes on fractal uncertainty principle
Lecture notes for lectures given at the Third Symposium on Scattering and Spectral Theory in Florianopolis, Brazil, July 2017 (work in progress).
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Lecture notes on scattering on convex co-compact hyperbolic surfaces
Lecture notes for lectures given in Tsinghua University in Summer 2016.
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Lecture notes on quantum ergodicity
Lecture notes for part of the course 18.158 given at MIT in
Spring 2016.
- Semiclassical Lagrangian distributions
Some (rather sloppy) notes for a (rather informal) talk on semiclassical Lagrangian distributions,
Fourier integral operators, and their applications to quantization conditions given
in March 2012.
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Propagation of singularities and nontrapping estimates
Student Harmonic Analysis and Differential Equations Seminar,
UC Berkeley, November 2011.
This is a short introduction to nontrapping estimates in scattering theory.
We discuss
(1) how one can obtain exponential decay in obstacle scattering
from a nontrapping estimate via the contour deformation argument
(2) how to prove the semiclassical propagation of singularities estimate
in the presence of a complex absorbing potential via Hörmander's
positive commutator method
(3) how propagation of singularities and complex scaling lead to a nontrapping
estimate in the one-dimensional model case.
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Pseudodifferential operators and elliptic regularity
Student Topology Seminar,
UC Berkeley, April–May 2010.
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Hörmander–Kashiwara and Maslov Indices
An article
that served as the final project in Michael Hutchings'
course on symplectic geometry in Spring 2009.
