arXiv subjects

Maciej Zworski

Publications and source records attributed to Maciej Zworski.

At least 19 recordsLinked to original sources

arXiv2026

Bouncing-ball modes in ergodic billiards

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian--Oliffson Kamphorst--Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

math.SP↗

arXiv2026

Optimal relaxation for Witten Lindbladians

We prove optimal trace-norm relaxation for the one-dimensional Lindbladian associated with the Witten differential $a=h\partial_x+V'$ which annihilates the classical Gibbs density: if $λ_1(h)$ is the first positive eigenvalue of $H=a^*a$, then the relaxation rate is $γ_h=λ_1(h)/(2h)$. It applies to initial operators whose Schwartz kernels, after a Gibbs conjugation, satisfy $ L^2 $ estimates for the restriction to the diagonal and for the normal derivative. An appendix by Chat GPT 6 presents a stronger result specialised to positive initial data.

math-ph↗

arXiv2026

Hartree--Fock coercivity for twisted bilayer graphene

We study the minimisers of the Hartree--Fock functional for flat bands in the chiral model of twisted bilayer graphene introduced by Tarnopolsky, Kruchkov, and Vishwanath. Our model is a continuous version of the discrete model considered by Becker, Lin and Stubbs in previous works and the first result is a direct analogue of main theorems of those papers: at half filling of the flat band of multiplicity two, there are exactly two minimisers, obtained by occupying either one of the two sub-bands. (A more general version of this result, stressing its topological origins, was presented in Appendix A.8 of the Lecture Notes by Tao-Zworski.) The new contribution is the strict coercivity of the Hessian of the Hartree--Fock functional at the two minimisers. It is a consequence of the difference between the Chern numbers of the component bands and can be optimistically considered as a soft version of a ``spectral gap''.

math-ph↗

arXiv2026

A mathematical study of periodic band inversion

We give a mathematical analysis of the periodic band inversion phenomenon observed by Tan--Devakul for an electron in a two-dimensional periodic potential coupled to a circularly polarized photon cavity mode. In the strong-coupling limit, we derive an effective Bloch Hamiltonian and prove convergence of the low-lying bands. For a cosine potential, we explain the periodic closing and reopening of the first spectral gap, prove the existence and generic persistence of Dirac cones at the gap-closing points, and compute the Chern numbers associated to isolated band clusters. We also show that higher isolated band clusters cannot persist in the small-coupling regime. Finally, we resolve an apparent sign discrepancy between Berry curvature computations and Chern numbers by tracking the descent from the covering space to the Brillouin torus.

math-ph↗

arXiv2026

Localised Davies generators for (pseudo)differential operators

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyén proposed a construction involving localisation in time and carried out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad evolution is expected to settle to the Gibbs state. In this paper, we show that the localised Davies construction also works for natural classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution. Our emphasis is microlocal: we prove that the localised jump operators are themselves pseudodifferential, and hence pseudolocal. The proof involves a novel version of Egorov's theorem valid for all times.

math-ph↗

arXiv2026

WKB structure in a scalar model of flat bands

We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods.

math-ph↗

arXiv2026

1D Scattering through time dependent media with memory

We construct a scattering matrix with operator valued entries describing solutions to the 1+1 wave equation where permittivities has memory and depends on time and space. It is the analogue of the scattering matrix for spatially localised perturbations where the entries are functions of frequency and appear as Fourier multipliers in solutions of the wave equation. This provides a mathematical explanation of the numerical construction in the recent paper by Horsley et al. The appendix by Zhen Huang and Maciej Zworski presents a numerical scheme for solving the wave equation considered in this article.

math.AP↗

arXiv2025

Overdamped QNM for Schwarzschild black holes

We prove that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild-de Sitter black holes in a disc of radius $ r $ is bounded from below by $ c r^3 $. This shows that the recent upper bound by Jézéquel is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM and explains the emergence of a distorted lattice on which they lie. Our presentation gives a general result about exponentially accurate Bohr-Sommerfeld quantization rules for one dimensional problems. The description of QNM allows their accurate evaluation ``deep in the complex" where numerical methods break down due to pseudospectral effects.

math-ph↗

arXiv2025

Degenerate flat bands in twisted bilayer graphene

We prove that in the chiral limit of the Bistritzer--MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be $ -1 $, and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer--MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials flat bands have only twofold or fourfold multiplicities.

math-ph↗

arXiv2024

Classical-Quantum correspondence in Lindblad evolution

We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into jump operators assumed to satisfy certain ellipticity conditions and modeling interaction with a larger system), the evolution of a quantum observable remains close to the classical Fokker--Planck evolution in the Hilbert--Schmidt norm for times vastly exceeding the Ehrenfest time (the limit of such agreement with no jump operators). The time scale is the same as in the recent papers by Hernández--Ranard--Riedel but the statement and methods are different. The appendix presents numerical experiments illustrating the classical/quantum correspondence in Lindblad evolution and comparing it to the mathematical results.

math-ph↗

arXiv2024

Discrete vs. continuous in the semiclassical limit

We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr-Sommerfeld quantization conditions for potentials exhibiting non-degenerate wells, and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter $h$ and show that it can be discontinuous.

math.SP↗

arXiv2024

Optimal enhanced dissipation for contact Anosov flows

We show that for a contact Anosov flow on a compact manifold $ M $, the solutions to $ \partial_t u + X u = νΔu $, $ ν> 0 $, where $ X $ is the generator of the flow and $ Δ$, a (negative) Laplacian for some Riemannian metric on $ M $, satisfy \[ \| u ( t ) - \underline u \|_{L^2 ( M) } \leq C ν^{-K} e^{ - βt } \| u( 0 ) \|_{L^2 ( M) }, \] where $ \underline u $ is the (conserved) average of $ u (0) $ with respect to the contact volume form, and $K$, $β$ are fixed positive constants. Since our class of flows includes geodesic flows on manifolds of negative curvature, this provides many examples of very precise optimal enhanced dissipation in the sense of [arXiv:1911.01561] and [arXiv:2304.05374]. The proof is based on results about stochastic stability of Pollicott--Ruelle resonances [arXiv:1407.8531].

math.AP↗

arXiv2023

Classically forbidden regions in the chiral model of twisted bilayer graphene. With an appendix by Zhongkai Tao and Maciej Zworski

We establish exponential decay, as the angle of twisting goes to $ 0$, of eigenstates in a model of twisted bilayer graphene (TBG), near the hexagon connecting stacking points. That is done by adapting microlocal methods Kawai-Kashiwara and Sjöstrand used to establish analytic hypoellipticity by Trépreau and Himonas. That replaces ellipticity, absent here, which is the usual mechanism behind classically forbidden regions. We also discuss numerical evidence of exponential decay in other regions (the center of the hexagon) and analytic complications involved in establishing that decay.

math-ph↗

arXiv2023

Mathematical results on the chiral model of twisted bilayer graphene (with an appendix by Mengxuan Yang and Zhongkai Tao)

The study of twisted bilayer graphene (TBG) is a hot topic in condensed matter physics with special focus on {\em magic angles} of twisting at which TBG acquires unusual properties. Mathematically, topologically non-trivial flat bands appear at those special angles. The chiral model of TBG pioneered by Tarnopolsky--Kruchkov--Vishwanath has particularly nice mathematical properties and we survey, and in some cases, clarify, recent rigorous results which exploit them.

cond-mat.mes-hall↗

arXiv2023

Mathematics of internal waves in a 2D aquarium

Following theoretical and experimental work of Maas et al we consider a linearized model for internal waves in effectively two dimensional aquaria. We provide a precise description of singular profiles appearing in long time wave evolution and associate them to classical attractors. That is done by microlocal analysis of the spectral Poincaré problem, leading in particular to a limiting absorption principle. Some aspects of the paper (for instance Section 6) can be considered as a natural microlocal continuation of the work of John on the Dirichlet problem for hyperbolic equations in two dimensions.

math.AP↗

arXiv2023

An abstract formulation of the flat band condition

Motivated by the study of flat bands in models of twisted bilayer graphene (TBG), we give abstract conditions which guarantee the existence of a discrete set of parameters for which periodic Hamiltonians exhibit flat bands. As an application, we show that a scalar operator derived from the chiral model of TBG has flat bands for a discrete set of parameters.

math.AP↗

arXiv2023

From the chiral model of TBG to the Bistritzer--MacDonald model

We analyse the splitting of exact flat bands in the chiral model of the twisted bilayer graphene (TBG) when the $AA'/BB'$ coupling of the full Bistritzer--MacDonald model is taken into account. The first-order perturbation caused by the $AA'/BB'$ potential the same for both bands and satisfies interesting symmetries, in particular it vanishes on the line defined by the $K$ points. The splitting of the flat bands is governed by the quadratic term which vanishes at the $K$ points.

math-ph↗

arXiv2023

Dirac points for twisted bilayer graphene with in-plane magnetic field

We study Dirac points of the chiral model of twisted bilayer graphene (TBG) with constant in-plane magnetic field. For a fixed small magnetic field, we show that as the angle of twisting varies between magic angles, the Dirac points move between $ K, K' $ points and the $ Γ$ point. The Dirac points for zero magnetic field and non magic angles lie at $ K $ and $ K'$, while in the presence of a non-zero magnetic field and near magic angles, they lie near the $ Γ$ point. For special directions of the magnetic field, we show that the Dirac points move, as the twisting angle varies, along straight lines and bifurcate orthogonally at distinguished points. At the bifurcation points, the linear dispersion relation of the merging Dirac points disappears and exhibit a quadratic band crossing point (QBCP). The results are illustrated by links to animations suggesting interesting additional structure.

math-ph↗