Annales Henri Poincaré最新文献

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.

We present a uniform (and unambiguous) procedure for scaling the matter fields in implementing the conformal method to parameterize and construct solutions of Einstein constraint equations with coupled matter sources. The approach is based on a phase space representation of the spacetime matter fields after a careful (n+1) decomposition into spatial fields B and conjugate momenta (Pi _B), which are specified directly and are conformally invariant quantities. We show that if the Einstein-matter field theory is specified by a Lagrangian which is diffeomorphism invariant and involves no dependence on derivatives of the spacetime metric in the matter portion of the Lagrangian, then fixing B and (Pi _B) results in conformal constraint equations that, for constant-mean curvature initial data, semi-decouple just as they do for the vacuum Einstein conformal constraint equations. We prove this result by establishing a structural property of the Einstein momentum constraint that is independent of the conformal method: For an Einstein-matter field theory which satisfies the conditions just stated, if B and (Pi _B) satisfy the matter Euler–Lagrange equations, then (in suitable form) the right-hand side of the momentum constraint on each spatial slice depends only on B and (Pi _B) and is independent of the spacetime metric. We discuss the details of our construction in the special cases of the following models: Einstein–Maxwell-charged scalar field, Einstein–Proca, Einstein-perfect fluid, and Einstein–Maxwell-charged dust. In these examples we find that our technique gives a theoretical basis for scaling rules, such as those for electromagnetism, that have worked pragmatically in the past, but also generates new equations with advantageous features for perfect fluids that allow direct specification of total rest mass and total charge in any spatial region.

We investigate the asymptotic distribution of large eigenvalues of the asymmetric quantum Rabi model with an integer static bias. For this model, we consider a variant of the generalized rotating-wave approximation, corresponding to perturbations of double eigenvalues. Using this idea, we obtain a three-term asymptotic formula for the m-th eigenvalue with the remainder estimate (O(m^{-1/2}ln m)) when m tends to infinity.

In dimensions (d=1,2,3), the Laplacian can be perturbed by a point potential. In higher dimensions, the Laplacian with a point potential cannot be defined as a self-adjoint operator. However, for any dimension there exists a natural family of functions that can be interpreted as Green’s functions of the Laplacian with a spherically symmetric point potential. In dimensions 1, 2, 3, they are the integral kernels of the resolvent of well-defined self-adjoint operators. In higher dimensions, they are not even integral kernels of bounded operators. Their construction uses the so-called generalized integral, a concept going back to Riesz and Hadamard. We consider the Laplace(–Beltrami) operator on the Euclidean space, the hyperbolic space and the sphere in any dimension. We describe the corresponding Green’s functions, also perturbed by a point potential. We describe their limit as the scaled hyperbolic space and the scaled sphere approach the Euclidean space. Especially interesting is the behavior of positive eigenvalues of the spherical Laplacian, which undergo a shift proportional to a negative power of the radius of the sphere. We expect that in any dimension our constructions yield possible behaviors of the integral kernel of the resolvent of a perturbed Laplacian far from the support of the perturbation. Besides, they can be viewed as toy models illustrating various aspects of renormalization in quantum field theory, especially the point-splitting method and dimensional regularization.

We analyse symmetries of Bloch eigenfunctions at magic angles for the Tarnopolsky–Kruchkov–Vishwanath chiral model of the twisted bilayer graphene (TBG) following the framework introduced by Becker–Embree–Wittsten–Zworski. We show that vanishing of the first Bloch eigenvalue away from the Dirac points implies its vanishing at all momenta, that is, the existence of a flat band. We also show how the multiplicity of the flat band is related to the nodal set of the Bloch eigenfunctions. We conclude with two numerical observations about the structure of flat bands.

Two-dimensional maps with discontinuities are considered. It is shown that, in the presence of discontinuities, the essential spectrum of the transfer operator is large whenever it acts on a Banach space with norm that is stronger than (L^infty ) or (BV). Three classes of examples are introduced and studied, both expanding and partially expanding. In two dimensions, there is complication due to the geometry of the discontinuities, an issue not present in the one-dimensional case and which is explored in this work.

In this article, we analyze the Bistritzer–MacDonald model (also known as the continuum model) of twisted bilayer graphene with an additional external magnetic field. We provide an explicit semiclassical asymptotic expansion of the density of states (DOS) in the limit of strong magnetic fields. We find that unlike for magnetic Schrödinger operators, perturbation of the chiral potentials do not expand the Landau bands while perturbations by the anti-chiral potentials do. The explicit expansion of the DOS also enables us to study magnetic response properties such as magnetic oscillations which includes Shubnikov-de Haas and de Haas-van Alphen oscillations as well as the integer quantum Hall effect.