Annales Henri Poincaré最新文献

In this paper, we investigate the inviscid limit (nu rightarrow 0) for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus ({mathbb {T}}^2), with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order (O(nu ^2)) and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.

We introduce supergroup analogs of 3-manifold invariants ({widehat{Z}}), also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the ({widehat{Z}}) invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup.

We consider the Schrödinger operator with regular short range complex-valued potential in dimension (dge 1). We show that, for (dge 2), the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for (d=1), we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for (d=3). Some directions for further research are formulated.

The twisted Ruelle zeta function of a contact, Anosov vector field, is shown to be equal, as a meromorphic function of the complex parameter (hbar in mathbb {C}) and up to a phase, to the partition function of an (hbar )-linear quadratic perturbation of BF theory, using an “axial” gauge fixing condition given by the Anosov vector field. Equivalently, it is also obtained as the expectation value of the same quadratic, (hbar )-linear, perturbation, within a perturbative quantisation scheme for BF theory, suitably generalised to work when propagators have distributional kernels.

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough–Dodd, the Federbush, and the O(n)-nonlinear sigma models.

Jackiw–Teitelboim dilaton quantum gravity localizes on a double-scaled random-matrix model, whose perturbative free energy is an asymptotic series. Understanding the resurgent properties of this asymptotic series, including its completion into a full transseries, requires understanding the nonperturbative instanton sectors of the matrix model for Jackiw–Teitelboim gravity. The present work addresses this question by setting-up instanton calculus associated with eigenvalue tunneling (or ZZ-brane contributions), directly in the matrix model. In order to systematize such calculations, a nonperturbative extension of the topological recursion formalism is required—which is herein both constructed and applied to the present problem. Large-order tests of the perturbative genus expansion validate the resurgent nature of Jackiw–Teitelboim gravity, both for its free energy and for its (multi-resolvent) correlation functions. Both ZZ and FZZT nonperturbative effects are required by resurgence, and they further display resonance upon the Borel plane. Finally, the resurgence properties of the multi-resolvent correlation functions yield new and improved resurgence formulae for the large-genus growth of Weil–Petersson volumes.

We consider a translation-invariant Pauli–Fierz model describing a non-relativistic charged quantum mechanical particle interacting with the quantized electromagnetic field. The charged particle may be spinless or have spin one half. We decompose the Hamiltonian with respect to the total momentum into a direct integral of so-called fiber Hamiltonians. We perform an infrared renormalization, in the sense of norm resolvent convergence, for each fiber Hamiltonian, which has the physical interpretation of removing an infinite photon cloud. We show that the renormalized fiber Hamiltonians have a ground state for almost all values for the total momentum with modulus less than one.

In this article, we define a generalization of the domino shuffling algorithm for tilings of the Aztec diamond to the interacting k-tilings recently introduced by S. Corteel, A. Gitlin, and the first author. We describe the algorithm both in terms of dynamics on a system of colored particles and as operations on the dominos themselves.

In the previous paper, we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation, we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform.

We study the solution of the two-temperature Fokker–Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular, we consider a spin glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra’s hierarchical construction appearing in Talagrand’s proof of the Parisi formula.