Annales Henri Poincaré最新文献

We study the dynamics of interacting fermions in the continuum. Our approach uses the concept of lattice-localized frames, which we introduce here. We first prove a Lieb-Robinson bound that is valid for a general class of local interactions, which implies the existence of the dynamics at the level of the CAR algebra. We then turn to the physical situation relevant to the (fractional) quantum Hall effect, namely the quasi-free second quantized Landau Hamiltonian to which electron–electron interactions can be added.

We consider a tight-binding model recently introduced by Timmel and Mele (Phys Rev Lett 125:166803, 2020) for strained moiré heterostructures. We consider two honeycomb lattices to which layer antisymmetric shear strain is applied to periodically modulate the tunneling between the lattices in one distinguished direction. This effectively reduces the model to one spatial dimension and makes it amenable to the theory of matrix-valued quasi-periodic operators. We then study the charge transport and spectral properties of this system, explaining the appearance of a Hofstadter-type butterfly and the occurrence of metal/insulator transitions that have recently been experimentally verified for non-interacting moiré systems (Wang et al. in Nature 577:42–46, 2020). For sufficiently incommensurable moiré lengths, described by a diophantine condition, as well as strong coupling between the lattices, which can be tuned by applying physical pressure, this leads to the occurrence of localization phenomena.

We give a functional characterization of a class of quasi-invariant determinantal processes corresponding to projection kernels in terms of de Branges spaces of entire functions.

Starting from the Hawking–Page solutions of [14], we consider the corresponding Lorentzian cone metrics. These represent cone interior scale-invariant vacuum solutions, defined in the chronological past of the scaling origin. We extend the Lorentzian Hawking–Page solutions to the cone exterior region in the class of ((4+1))-dimensional scale-invariant vacuum solutions with an (SO(3)times U(1)) isometry, using the Kaluza–Klein reduction and the methods of Christodoulou in [5]. We prove that each Lorentzian Hawking–Page solution has extensions with a null curvature singularity, extensions with a spacelike curvature singularity, and extensions with a null Cauchy horizon of Taub–NUT type. These are all the possible extensions within our symmetry class. The extensions to spacetimes with a null curvature singularity can be used to construct ((4+1))-dimensional asymptotically flat vacuum spacetimes with locally naked singularities, where the null curvature singularity is not preceded by trapped surfaces. We prove the instability of such locally naked singularities using the blue-shift effect of Christodoulou in [6].

By extending the gauge covariant magnetic perturbation theory to operators defined on half-planes, we prove that for 2d random ergodic magnetic Schrödinger operators, the zero-temperature bulk-edge correspondence can be obtained from a general bulk-edge duality at positive temperature involving the bulk magnetization and the total edge current. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a “zero-temperature” limit.

We study the large field limit in Schrödinger equations with magnetic vector potentials describing translationally invariant B-fields with respect to the z-axis. In a first step, using regular perturbation theory, we derive an approximate description of the solution, provided the initial data are compactly supported in the Fourier variable dual to (zin {{mathbb {R}}}). The effective dynamics is thereby seen to produce high-frequency oscillations and large magnetic drifts. In a second step, we show, by using the theory of almost invariant subspaces, that this asymptotic description is stable under polynomially bounded perturbations that vanish in the vicinity of the origin.

We study the ground state energy of a gas of N fermions confined to a unit box in d dimensions. The particles interact through a two-body potential with strength scaled in an N-dependent way as (N^{-alpha }v), where (alpha in mathbb {R}) and v is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case (alpha <1-frac{2}{d}). We contrast our result with existing results in the weakly interacting case (alpha >1-frac{2}{d}) and the transition happening at the mean-field scaling (alpha =1-frac{2}{d}). Our proof is an adaptation of the bosonization technique used to treat the mean-field case.

We obtain dispersive and Strichartz estimates for solutions to the Schrödinger equation with one Aharonov–Bohm solenoid in the uniform magnetic field. The main step of the proof is the construction of the Schrödinger kernel, and the main obstacle is to obtain the explicit representation of the kernel, which requires a large set of careful calculations. To overcome this obstacle, we plan to construct the Schrödinger kernel by two different strategies. The first one is to use the Poisson summation formula as Fanelli et al. (Adv Math 400:108333, 2022), while the second one relies on the Schulman–Sunada formula in Št’ovíček (Ann Phys 376:254–282, 2017), which reveals the intrinsic connections of the heat kernels on manifolds with group actions.

We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. Here, the asymptotics refer to the behavior of the solutions near a cosmological singularity, or near infinity in the expanding direction. Leading-order asymptotics for solutions of silent equations were already obtained by Ringström (Astérisque 420, 2020). Here, we improve upon Ringström’s result, by obtaining asymptotic estimates of all orders for the solutions, and showing that solutions are uniquely determined by the asymptotic data contained in the estimates. As an application, we then study solutions to the source free Maxwell’s equations in Kasner spacetimes near the initial singularity. Our results allow us to obtain an asymptotic expansion for the potential of the electromagnetic field, and to show that the energy density of generic solutions blows up along generic timelike geodesics when approaching the singularity. The asymptotics we study correspond to the heuristics of the BKL conjecture, where the coefficients of the spatial derivative terms of the equations are expected to be small, and thus these terms are neglected in order to obtain the asymptotics.