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In this paper we study a so-called separatrix map introduced by Zaslavskii-Filonenko (Sov Phys JETP 27:851-857, 1968) and studied by Treschev (Physica D 116(1-2):21-43, 1998; J Nonlinear Sci 12(1):27-58, 2002), Piftankin (Nonlinearity (19):2617-2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3-108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at , 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at , 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior. [ABSTRACT FROM AUTHOR]

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators Ph, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that ( Ph − I)/ h converges for small h to a second order elliptic differential operator $${\mathcal{L}}$$ L on compactly supported functions and that the Markov chain process associated to Ph converges to a diffusion with infinitesimal generator $${\mathcal{L}}$$ L . Both Ph and $${\mathcal{L}}$$ L are self-adjoint (densely) defined on the space $${L^2(\mathbb{H},\eta)}$$ L 2 ( H , η ) of square-integrable functions over the (lower) half-space $${\mathbb{H}}$$ H in $${\mathbb{R}^m}$$ R m , where η is a stationary measure. This measure’s density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator $${\mathcal{L}}$$ L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes. [ABSTRACT FROM AUTHOR]

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes' continued fractions and multi-point Padé approximations.We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds, as well as Toda-like sorting property. [ABSTRACT FROM AUTHOR]

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the L norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of strong or weak solutions of the Navier-Stokes equations in the case of an homogeneous initial data. The case of small initial data is discussed. For large initial data, an approximation is developed, in the spirit of a paper of Vishik and Fursikov. Qualitative convergence is obtained by use of the theory of Muckenhoupt weights. [ABSTRACT FROM AUTHOR]

The Aharonov-Bohm effect is a fundamental issue in physics that has been extensively studied in the literature and is discussed in most of the textbooks in quantum mechanics. The issues at stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. The Aharonov-Bohm effect is a very controversial issue. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. (Phys Rev Lett 48:1443-1446, ; Phys Rev Lett 56:792-795, ) with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. (Phys Rev Lett 99:210401, ) that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. Aharonov and Bohm (Phys Rev 115:485-491, ) proposed an Ansatz for the solution to the Schrödinger equation in simply connected regions of space where there are no electromagnetic fields. It consists of multiplying the free evolution by the Dirac magnetic factor. The Aharonov-Bohm Ansatz predicts the results of the experiments of Tonomura et al. and of Caprez et al. Recently in Ballesteros and Weder (Math Phys 50:122108, ) we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al. We provided a rigorous, simple, quantitative, error bound for the difference in norm between the exact solution and the Aharonov-Bohm Ansatz. In this paper we prove that these results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handlebodies. Each handlebody is diffeomorphic to a torus or a ball, and some of them can be patched though the boundary. We formulate the Aharonov-Bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by v, 0 < ρ < 1, with v the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature. Namely, that magnetic fields act at a distance on charged particles, and that this action at a distance is carried by the circulation of the magnetic potential which gives a real physical significance to magnetic potentials. [ABSTRACT FROM AUTHOR]

Stein’s method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson’s circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold. [ABSTRACT FROM AUTHOR]

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by $${2\frac{kd^2}{n}}$$ , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation $${U_{\mu +\nu} \subset U_\mu \otimes U_\nu}$$ of the unitary group U( d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out U ν. Then ξμ is close to a convex combination of the coherent states $${U_\mu(g)|{v_\mu\rangle}}$$ , where $${g\in U(d)}$$ and $${|v_\mu\rangle}$$ is the highest weight vector in U μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general. [ABSTRACT FROM AUTHOR]

In this paper we analyze the behavior of the solution of the dissipative Boussinesq systems where α, β, c > 0 are parameters. Those systems model two-dimensional small amplitude long wavelength water waves. For α ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β, c and almost every α ≤ 1, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form. [ABSTRACT FROM AUTHOR]

In this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized. [ABSTRACT FROM AUTHOR]

We revisit the diffusive limit of a steady neutron transport equation in a two-dimensional unit disk with one-speed velocity. A classical theorem by Bensoussan et al. (Publ Res Inst Math Sci 15(1):53-157, ) states that its solution can be approximated in L by the leading order interior solution plus the Knudsen layer in the diffusive limit. In this paper, we construct a counterexample to this result via a different boundary layer expansion with geometric correction. [ABSTRACT FROM AUTHOR]

Non-periodic tilings and local rules are commonly used to model the long range aperiodic order of quasicrystals and the finite-range energetic interactions that stabilize them. This paper focuses on planar rhombus tilings, which are tilings of the Euclidean plane, which can be seen as an approximation of a real plane embedded in a higher dimensional space. Our main result is a characterization of the existence of local rules for such tilings when the embedding space is four-dimensional. The proof is an interplay of algebra and geometry that makes use of the rational dependencies between the coordinates of the embedded plane. We also apply this result to some cases in a higher dimensional embedding space, notably tilings with n-fold rotational symmetry. [ABSTRACT FROM AUTHOR]

We consider modulational solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem, neglecting surface tension. For such solutions, it is well known that one formally expects the modulation to be a profile traveling at group velocity and governed by a 2D hyperbolic cubic nonlinear Schrödinger equation. In this paper we justify this fact by providing rigorous error estimates in Sobolev spaces. We reproduce the multiscale calculation to derive an approximate wave packet-like solution to the evolution equations with mild quadratic nonlinearities constructed by Sijue Wu. Then we use the energy method along with the method of normal forms to provide suitable a priori bounds on the difference between the true and approximate solutions. [ABSTRACT FROM AUTHOR]

In this paper, we give a rigorous derivation of Einstein's geodesic hypothesis in general relativity. We use small material bodies $${\phi^\epsilon}$$ governed by the nonlinear Klein-Gordon equations to approximate the test particle. Given a vacuum spacetime $${([0, T]\times\mathbb{R}^3, h)}$$ , we consider the initial value problem for the Einstein-scalar field system. For all sufficiently small ε and δ ≤ ε, q > 1, where δ, ε are the amplitude and size of the particle, we show the existence of the solution $${([0, T]\times\mathbb{R}^3, g, \phi^\epsilon)}$$ to the Einstein-scalar field system with the property that the energy of the particle $${\phi^\epsilon}$$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by $${\phi^\epsilon}$$ is negligibly small in C, that is, the spacetime metric g is C close to the given vacuum metric h. These results generalize those obtained by Stuart in (Ann Sci École Norm Sup (4) 37(2):312-362, , J Math Pures Appl (9) 83(5):541-587, ). [ABSTRACT FROM AUTHOR]

It is well known that the low-energy classical dynamics of solitons of Bogomol'nyi type is well approximated by geodesic motion in $${{\sf M}_n}$$, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction $${\psi : {\sf M}_n \rightarrow \mathbb{C}}$$ evolves according to the Hamiltonian $${H_0 = \frac{1}{2} \triangle}$$, where $${\triangle}$$ is the Laplacian on $${{\sf M}_n}$$ . Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including $${\kappa}$$, the scalar curvature of $${{\sf M}_n}$$, and $${\fancyscript{C}}$$, the n-soliton Casimir energy, which are usually difficult or impossible to compute, and whose effect on the energy spectrum is unknown. This paper analyzes the spectra of H and two corrections to it suggested by work of Moss and Shiiki, namely $${H_1 = H_0 + \frac{1}{4} \kappa}$$ and $${H_2 = H_1 + \fancyscript{C}}$$, in the simple but nontrivial case of a single $${\mathbb{C}P^1}$$ lump moving on the two-sphere. Here $${{\sf M}_1 = {\sf Rat}_1}$$, a noncompact kähler 6-manifold invariant under an $${SO(3)\times SO(3)}$$ action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. By exploiting the diffeomorphism $${{\sf Rat}_1\cong TSO(3)}$$, a hidden isometry of $${{\sf Rat}_1}$$ is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on an SO(3) submanifold of $${{\sf Rat}_1}$$, using standard results from harmonic map theory and zeta function regularization, and approximated numerically on the rest of $${{\sf Rat}_1}$$. The lowest 19 eigenvalues of H are found, and their spin-isospin and parity compared for i = 0, 1, 2. It is found that the curvature corrections in H lead to a qualitatively unchanged low-level spectrum while the Casimir energy in H leads to significant changes. The scaling behaviour of the spectra under changes in the radii of the domain and target spheres is analyzed, and it is found that the disparity between the spectra of H and H is reduced when the target sphere is made smaller. [ABSTRACT FROM AUTHOR]

We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system. [ABSTRACT FROM AUTHOR]

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $${\varepsilon \ll 1}$$ controls the separation of time scales and the limit $${\varepsilon\to 0}$$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $${\varepsilon\to 0}$$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $${\varepsilon}$$ -dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn (Phys Rev A (3) 44(8):5239-5256, ), coming from an $${\varepsilon}$$ -dependent classical Hamilton function and an $${\varepsilon}$$ -dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $${\varepsilon^2}$$ . In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics (Xiao et al. in Rev Mod Phys 82(3):1959-2007, ). Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation. [ABSTRACT FROM AUTHOR]

We consider a construction of a family of almost-sharp fronts for the Surface Quasi-Geostrophic equation. The family is indexed by a parameter indicating the thickness of the almost-sharp front. In this paper we obtain the limit equation when the parameter approaches zero and construct approximate solutions of SQG in the analytic class. [ABSTRACT FROM AUTHOR]

In this paper, we consider families of operators $${\{x_r\}_{r \in \Lambda}}$$ in a tracial C*-probability space $${({\mathcal{A}}, \varphi)}$$ , whose joint *-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $${\{H_n^+\}_{n \in \mathbb {N}}}$$ . We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $${{\mathcal{B}}}$$ generated by $${\{x_r\}_{r \in \Lambda}}$$ , which generalizes the strong Haagerup inequalities for *-free R-diagonal families obtained by Kemp-Speicher (J Funct Anal 251:141-173, ). As an application of our result, we show that $${{\mathcal{B}}}$$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C*-algebra of the free unitary quantum group $${U_n^+}$$ . We show that the non-self-adjoint subalgebra $${{\mathcal{B}}_n}$$ generated by the matrix elements of the fundamental corepresentation of $${U_n^+}$$ has the MAP. Additionally, we prove a strong Haagerup inequality for $${{\mathcal{B}}_n}$$ , which improves on the estimates given by Vergnioux's property RD (Vergnioux in J Oper Theory 57:303-324, ). [ABSTRACT FROM AUTHOR]

In this paper the effective mass approximation and the k·p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period $${\epsilon}$$ of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k·p and effective mass models, well known in solid-state physics, are proved to be close (in the strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation. [ABSTRACT FROM AUTHOR]

In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291(1):257-302 ()], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper's main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a class of C1-smooth approximate solutions { uε, pε} to the 3D steady axisymmetric Euler equations will converge strongly to 0 in $${L^2_{loc}(R^3)}$$ . The main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields. We also show a Liouville type theorem that there are no non-trivial C1-smooth exact solutions with finite energy and uniform constant state at far fields. [ABSTRACT FROM AUTHOR]

Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples. [ABSTRACT FROM AUTHOR]

Let be a sequence of skew-symmetric polynomials in X1,⋯,Xl satisfying degXjPn,l≤n-1, whose coefficients are symmetric Laurent polynomials in z1,⋯,zn. We call p an ∞-cycle if holds for all n,l. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables αa=-logXa are the integration variables and βj=logzj are the rapidity variables. To each ∞-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞-cycles. In this paper, we define an action of on the space of ∞-cycles. There are two sectors of ∞-cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞-cycles which are polynomials in z1,⋯,zn is equal to the level (-1) irreducible character of with lowest weight -Λ0 (resp. -Λ1). We also suggest a possible tensor product structure of the full space of ∞-cycles. [ABSTRACT FROM AUTHOR]

: We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions. [ABSTRACT FROM AUTHOR]

We determine the density of eigenvalues of the scattering matrix of the Schrödinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential. [ABSTRACT FROM AUTHOR]

We have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE Trans. IT 46, 810–819 (2009)), the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning ‘approximation error’ up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression. [ABSTRACT FROM AUTHOR]

We consider the time evolution of a disk under the action of a constant force and interacting with a free gas in the mean-field approximation. Letting V 0>0 be the initial velocity of the disk and V ∞>0 its equilibrium velocity, namely the one for which the external field is balanced by the friction force exerted by the background, we show that, if V ∞− V 0 is positive and sufficiently small, then the disk reaches V ∞ with the power law t − ( d+2), d=1,2,3 being the dimension of the physical space. The reason for this behavior is the long tail memory due to recollisions. Any Markovian approximation (or simply neglecting the recollisions) yields an exponential approach to equilibrium. [ABSTRACT FROM AUTHOR]

The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories. [ABSTRACT FROM AUTHOR]

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-calledBethe approximation. Specifically, we show that the spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on ann-vertex regular tree with (+)-boundary are bounded below by a constant independent ofnat all temperatures and all external fields. This implies that the mixing time isO(logn) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard-core lattice gas (independent sets). [ABSTRACT FROM AUTHOR]

: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site. [ABSTRACT FROM AUTHOR]

We show that the Hawking-Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of C1,1-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for C1,1-metrics, and of C0-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor, have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have been recently shown to be rare, in the sense that for “most” proportionality factors there are no nontrivial Beltrami fields of high enough regularity (e.g., of class C6,α), not even locally. Our objective in this work is to show that, nevertheless, there are “many” Beltrami fields with non-constant factor, even realizing arbitrarily complicated vortex structures. This fact is relevant in the study of turbulent configurations. The core results are an “almost global” stability theorem for strong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinity can be perturbed to get “many” Beltrami fields with non-constant factor of arbitrarily high regularity and defined in the exterior of an arbitrarily small ball, and a “local” stability theorem for generalized Beltrami fields, which is an analogous perturbative result which is valid for any kind of Beltrami field (not just with a constant factor) but only applies to small enough domains. The proof relies on an iterative scheme of Grad-Rubin type. For this purpose, we study the Neumann problem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equation method and we obtain Hölder estimates, a sharp decay at infinity and some compactness properties for these sequences of approximate solutions. Some of the parts of the proof are of independent interest. [ABSTRACT FROM AUTHOR]

We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit ∫V→8πa, we determine the critical temperature to be Tc=Tfc(1+1.49ρ1/3a) to leading order. Here, Tfc is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee-Huang-Yang formula in the limit ∫V→8πa. [ABSTRACT FROM AUTHOR]

We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of n fermionic modes and has a Hamiltonian $${H=H_0+H_{imp}}$$ , where H is quadratic in creation-annihilation operators and H is an arbitrary Hamiltonian acting on a subset of O(1) modes. We show that the ground energy of H can be approximated with an additive error $${2^{-b}}$$ in time $${n^3 \exp{[O(b^3)]}}$$ . Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of $${\exp{[O(b^3)]}}$$ fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of H . A key ingredient of our proof is Zolotarev's rational approximation to the $${\sqrt{x}}$$ function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model. [ABSTRACT FROM AUTHOR]

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014) and the corresponding uniqueness result of Gubinelli and Perkowski (Energy solutions of KPZ are unique, 2015). [ABSTRACT FROM AUTHOR]

We study a system of M particles in contact with a large but finite reservoir of $${N \gg M}$$ particles within the framework of the Kac master equation modeling random collisions. The reservoir is initially in equilibrium at temperature $${T=\beta^{-1}}$$ . We show that for large N, this evolution can be approximated by an effective equation in which the reservoir is described by a Maxwellian thermostat at temperature T. This approximation is proven for a suitable $${L^2}$$ norm as well as for the Gabetta-Toscani-Wennberg (GTW) distance and is uniform in time. [ABSTRACT FROM AUTHOR]

We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in a rather strong sense; namely, the (normalized) Heisenberg evolution of the position operator converges strongly to a self-adjoint operator that is injective on the space of absolutely summable sequences. In particular, this means that all transport exponents corresponding to well-localized initial states are equal to one. Our result may be applied to a class of quantum many-body problems. Specifically, we establish a lower bound on the Lieb-Robinson velocity for an isotropic XY spin chain on the integers with limit-periodic couplings. [ABSTRACT FROM AUTHOR]

We consider two finite index endomorphisms $${\rho}$$ , $${\sigma}$$ of any AFD factor M. We characterize the condition for there being a sequence $${\{ u_n\}}$$ of unitaries of the factor M with $${\mathrm{Ad}u_n \circ \rho \to \sigma}$$ . The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of AFD factors, obtained by Kawahiashi-Sutherland-Takesaki and Masuda-Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras. [ABSTRACT FROM AUTHOR]

We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states. [ABSTRACT FROM AUTHOR]

A state on a tripartite quantum system $${A \otimes B \otimes C}$$ forms a Markov chain if it can be reconstructed from its marginal on $${A \otimes B}$$ by a quantum operation from B to $${B \otimes C}$$ . We show that the quantum conditional mutual information I( A : C| B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated. [ABSTRACT FROM AUTHOR]

We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein-Primakoff representation. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice. [ABSTRACT FROM AUTHOR]

For the Glimm scheme approximation $${u_\varepsilon}$$ to the solution of the system of conservation laws in one space dimension with initial data u with small total variation, we prove a quadratic (w.r.t. Tot. Var. ( u)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely, we obtain the following results: This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $${D_{t} \hat \sigma_k(t,w)}$$ is a measure with total mass $${\leq \mathcal{O}(1) {\rm Tot. Var.} (u_0)^2}$$ , where $${\hat{\sigma}_k(t, w)}$$ is the speed given to the wave w by the Riemann problem at the grid point $${(i\varepsilon, \mathtt{x}(i\varepsilon, w)), t \in [i\varepsilon, (i + 1)\varepsilon)}$$ . [ABSTRACT FROM AUTHOR]

We construct approximate transport maps for non-critical $${\beta}$$ -matrix models, that is, maps so that the push forward of a non-critical $${\beta}$$ -matrix model with a given potential is a non-critical $${\beta}$$ -matrix model with another potential, up to a small error in the total variation distance. One of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models. [ABSTRACT FROM AUTHOR]

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient. [ABSTRACT FROM AUTHOR]

The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state φ is shown to imply existence of unital, φ-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms. [ABSTRACT FROM AUTHOR]

We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions $${N \geq 2}$$ via the approach of transformation optics. There are four major ingredients in our proposed theory: (1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation that blows up a subset $${K_\varepsilon}$$ in the virtual space, where $${\varepsilon \ll 1}$$ is an asymptotic regularization parameter. $${K_\varepsilon}$$ will degenerate to K as $${\varepsilon \rightarrow +0}$$ , and in our theory K could be any convex compact set in $${\mathbb{R}^N}$$ , or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. (2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. (3) The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. (4) In order to achieve a cloaking device of compact size, particularly for the case when $${K_\varepsilon}$$ is not 'uniformly small', an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field $${u_\varepsilon}$$ corresponding to a cloaking problem will converge to u as $${\varepsilon \rightarrow +0}$$ , with u being the scattered wave field corresponding to a sound-hard K. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We formulate a generalized concept of asymptotic completeness and show that it holds in any Haag-Kastler quantum field theory with an upper and lower mass gap. It remains valid in the presence of pairs of oppositely charged particles in the vacuum sector, which invalidate the conventional property of asymptotic completeness. Our result can be restated as a criterion characterizing a class of theories with complete particle interpretation in the conventional sense. This criterion is formulated in terms of certain asymptotic observables (Araki-Haag detectors) whose existence, as strong limits of their approximating sequences, is our main technical result. It is proven with the help of a novel propagation estimate, which is also relevant to scattering theory of quantum mechanical dispersive systems. [ABSTRACT FROM AUTHOR]

We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra $${A(\mathbb{G})}$$ whose representing multipliers form an approximate identity in $${C_0(\mathbb{G})}$$ that is bounded in the $${M0A(\mathbb{G})}$$ norm; the bound being an upper estimate for the associated Cowling-Haagerup constant. As an application, we find the appropriate approximation properties of the extended quantum SU(1, 1) group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White. [ABSTRACT FROM AUTHOR]

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SU(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SU(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP. [ABSTRACT FROM AUTHOR]