Annales Henri Poincaré

List of Papers (Total 74)

Universality of Single-Qudit Gates

We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G$ is universal, i.e. if ${<}\mathcal {S}{>}$ is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in $\mathcal {S}$ we assign the orthogonal matrix $\mathrm {Ad}_g$ that is image of g under the adjoint ...

A Geometric Invariant Characterising Initial Data for the Kerr–Newman Spacetime

We describe the construction of a geometric invariant characterising initial data for the Kerr–Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr–Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr–Newman spacetime in terms of Killing spinors. The ...

Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic many-body models: part 1—main results and algebra, arXiv:​1609.​01745, 2016, The small field parabolic flow for ...

Homogeneous Rank One Perturbations

A holomorphic family of closed operators with a rank one perturbation given by the function $x^{\frac{m}{2}}$ is studied. The operators can be used in a toy model of renormalization group.

Small Cocycles, Fine Torus Fibrations, and a $\varvec{\mathbb {Z}^{2}}$ Subshift with Neither

Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free $\mathbb {Z}^d$ actions on Cantor sets admit “small cocycles.” These represent classes in $H^1$ that are mapped to small vectors in $\mathbb {R}^d$ by the Ruelle–Sullivan (RS) map. We show that there exist $\mathbb {Z}^2$ actions where no such ...

Decay of Correlations in 2D Quantum Systems with Continuous Symmetry

We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan–Spencer–Koma–Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg, Hubbard, and t-J models, and to certain models of random loops.

Causality for Nonlocal Phenomena

Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime ...

Tropical Amplitudes

In this work, we argue that the $\alpha '\rightarrow 0$ limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus. An important technical input from tropical geometry is the use of tropical theta functions with characteristics to ...

Lieb–Thirring Inequalities for Finite and Infinite Gap Jacobi Matrices

We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum.

Perturbations of the Asymptotic Region of the Schwarzschild–de Sitter Spacetime

The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the ...

Models for Self-Gravitating Photon Shells and Geons

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein–Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric ...

Spectral Theory for Schrödinger Operators with $\varvec{\delta }$ -Interactions Supported on Curves in $\varvec{\mathbb {R}^3}$

The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with $\delta$-interactions supported on closed curves in $\mathbb {R}^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive ...

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

The paper is devoted to operators given formally by the expression \begin{aligned} -\partial _x^2+\left( \alpha -\frac{1}{4}\right) \frac{1}{x^{2}}. \end{aligned}This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real $\alpha$, or closed operator for complex $\alpha$, we find that this homogeneity can be ...

Non-Backtracking Loop Soups and Statistical Mechanics on Spin Networks

We introduce and study a Markov field on the edges of a graph $\mathcal {G}$ in dimension $d\ge 2$ whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a ...

Asymptotics for Erdős–Solovej Zero Modes in Strong Fields

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on ${\mathbb{R}^3}$. In particular, we are interested in those operators ${\mathcal{D}_B}$ for which the associated magnetic field ${B}$ is given by pulling back a two-form ${\beta}$ from the sphere ${\mathbb{S}^2}$ to ${\mathbb{R}^3}$ using a combination of the ...

Elliptic Genera and 3d Gravity

We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of K3, product manifolds, certain simple families of Calabi–Yau hypersurfaces, ...

Dispersion Estimates for Spherical Schrödinger Equations

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

Integrable QFT and Longo–Witten Endomorphisms

Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left–left, right–right and left–right scattering functions. We find a correspondence between massless left–right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class ...

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of ...

Hypergeometric Type Functions and Their Symmetries

The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F 1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are ...

Infraparticle Problem, Asymptotic Fields and Haag–Ruelle Theory

In this article, we want to argue that an appropriate generalization of the Wigner concepts may lead to an asymptotic particle with well-defined mass, although no mass hyperboloid in the energy–momentum spectrum exists.