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Volume 53, Issue 9, September 2012
 SPECIAL ISSUE: IN HONOR OF ELLIOTT LIEB’S 80TH BIRTHDAY


Introduction to Special Issue: In Honor of Elliott Lieb's 80th birthday
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LiebThirring inequality for a model of particles with point interactions
View Description Hide DescriptionWe consider a model of quantummechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a LiebThirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power .

Relativistic Scott correction in selfgenerated magnetic fields
View Description Hide DescriptionWe consider a large neutral molecule with total nuclear charge Z in a model with selfgenerated classical magnetic field and where the kinetic energy of the electrons is treated relativistically. To ensure stability, we assume that Zα < 2/π, where α denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit Z → ∞, α → 0 such that κ = Zα is fixed. The leading term in the energy asymptotics is independent of κ, it is given by the ThomasFermi energy of order Z ^{7/3} and it is unchanged by including the selfgenerated magnetic field. We prove the first correction term to this energy, the socalled Scott correction of the form S(αZ)Z ^{2}. The current paper extends the result of Solovej et al. [Commun. Pure Appl. Math.LXIII, 39–118 (2010)] on the Scott correction for relativistic molecules to include a selfgenerated magnetic field. Furthermore, we show that the corresponding Scott correction function S, first identified by Solovej et al. [Commun. Pure Appl. Math.LXIII, 39–118 (2010)], is unchanged by including a magnetic field. We also prove new LiebThirring inequalities for the relativistic kinetic energy with magnetic fields.

Critical rotational speeds for superfluids in homogeneous traps
View Description Hide DescriptionWe present an asymptotic analysis of the effects of rapid rotation on the ground state properties of a superfluid confined in a twodimensional trap. The trapping potential is assumed to be radial and homogeneous of degree larger than two in addition to a quadratic term. Three critical rotational velocities are identified, marking, respectively, the first appearance of vortices, the creation of a “hole” of low density within a vortex lattice, and the emergence of a giant vortex state free of vortices in the bulk. These phenomena have previously been established rigorously for a “flat” trap with fixed boundary but the “soft” traps considered in the present paper exhibit some significant differences, in particular the giant vortex regime, that necessitate a new approach. These differences concern both the shape of the bulk profile and the size of vortices relative to the width of the annulus where the bulk of the superfluid resides. Close to the giant vortex transition the profile is of ThomasFermi type in “flat” traps, whereas it is gaussian for soft traps, and the “last” vortices to survive in the bulk before the giant vortex transition are small relative to the width of the annulus in the former case but of comparable size in the latter.

Symmetry of extremals of functional inequalities via spectral estimates for linear operators
View Description Hide DescriptionWe prove new symmetry results for the extremals of the CaffarelliKohnNirenberg inequalities in any dimension larger or equal than 2 , in a range of parameters for which no explicit results of symmetry were previously known.

Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs
View Description Hide DescriptionWe discuss the dynamical implications of the recent proof that for a quantum particle in a random potential on a regular tree graph absolutely continuous (ac) spectrum occurs nonperturbatively through rare fluctuationenabled resonances. The main result is spelled in the title.

Brownian motions on metric graphs
View Description Hide DescriptionBrownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.

Stability of impurities with Coulomb potential in graphene with homogeneous magnetic field
View Description Hide DescriptionGiven a twodimensional nopair Weyl operator W _{ Z } with a point nucleus of charge Z, we show that a homogeneous magnetic field does not lower the critical charge beyond which it collapses.

Persistent energy flow for a stochastic wave equation model in nonequilibrium statistical mechanics
View Description Hide DescriptionWe consider a onedimensional partial differential equation system modeling heat flow around a ring. The system includes a KleinGordon wave equation for a field satisfying spatial periodic boundary conditions, as well as OrnsteinUhlenbeck stochastic differential equations with finite rank dissipation and stochastic driving terms modeling heat baths. There is an energy flow around the ring. In the case of a linear field with different (fixed) bath temperatures, the energy flow can persist even when the interaction with the baths is turned off. A simple example is given.

Existence of the thermodynamic limit for disordered quantum Coulomb systems
View Description Hide DescriptionFollowing a recent method introduced by Hainzl, Solovej, and Lewin, we prove the existence of the thermodynamic limit for a system made of quantum electrons, and classical nuclei whose positions and charges are randomly perturbed in an ergodic fashion. All the particles interact through Coulomb forces.

Entanglement, Bell inequality and all that
View Description Hide DescriptionWe start from the geometrical observation that a finite set of pure states correspond to some points on a sphere and their convex span cannot be the whole set of states. If we call the left over entangled we can pursue this picture from the simplest case of a two dimensional Hilbert space to the usual AliceandBob game of entangled states and then move to bigger systems and finely to quantum field theory where almost everything is entangled. On the way we encounter more or less known old friends up from the shell structure of states to the monogamy of squashed entanglement. We study how entanglement can be concentrated on a small slice and how it depends on the particular factorization of the Hilbert space.

Location of the LeeYang zeros and absence of phase transitions in some Ising spin systems
View Description Hide DescriptionWe consider a class of Ising spin systems on a set Λ of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the Λ LeeYang zeros in the complex z = e ^{2βh } plane, where β is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of β. In some cases one obtains, in an appropriately taken β↗ ∞ limit, a gas of hard objects on a set Λ^{′}; the fugacity for the limiting system is a rescaling of z and the LeeYang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the LeeYang zeros of both the finite and zerotemperature systems lie on the negative real axis for all β. One zerotemperature limit of this type, for example, is a monomerdimer system; our results thus generalize, to finite β, a wellknown result of Heilmann and Lieb that the LeeYang zeros of monomerdimer systems are real and negative.

A contribution to the manyfermion problem
View Description Hide DescriptionIn this work we reexamine the manyfermion problem in arbitrary dimensions. It is shown that in two dimensions or higher, the Hamiltonian of interacting fermions can be separated into individual nonintersecting sectors labeled by the wavevector . Within each sector the Hamiltonian maps onto a generalized version of the onedimensional Luttinger model that resembles a boson string. These are chainlike quadratic forms in boson operators that are readily diagonalized in the absence of “exchange” corrections. Moreover, in a simple example involving SU(2) fermions, that of the Hubbard model, we show that it can be possible also to incorporate exchange terms and express them entirely within an enlarged set of string variables.

Indirect Coulomb energy for twodimensional atoms
View Description Hide DescriptionIn this paper we provide a family of lower bounds on the indirect Coulomb energy for atomic and molecular systems in two dimensions in terms of a functional of the single particle density with gradient correction terms.

The scaling limit of the energy correlations in nonintegrable Ising models
View Description Hide DescriptionWe obtain an explicit expression for the multipoint energy correlations of a nonsolvable twodimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength λ, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis, and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in λ. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.

Scale dependence of the retarded van der Waals potential
View Description Hide DescriptionWe study the ground stateenergy for a system of two hydrogen atoms coupled to the quantized Maxwell field in the limit α → 0 together with the relative distance between the atoms increasing as α^{−γ} R, γ > 0. In particular we determine explicitly the crossover function from the R ^{−6} van der Waals potential to the R ^{−7} retarded van der Waals potential, which takes place at scale α^{−2} R.

Fluctuation estimates for subquadratic gradient field actions
View Description Hide DescriptionIn this article we estimate fluctuations of the scalar field ϕ for a special class of subquadratic actions which grow like ∇ϕ^{2α}, 0 < α < 1. In particular if α = 1/2 we show that in three dimensions is bounded for γ small. For each edge (jk) we introduce an auxiliary field to express the action as a superposition of Gaussian free fields. The effective action which arises from integrating over the Gaussian field is shown to be convex in t. The BrascampLieb inequality is then applied to obtain the desired estimates on a nonuniformly elliptic Green's function.

Symplectic cohomologies on phase space
View Description Hide DescriptionThe phase space of a particle or a mechanical system contains an intrinsic symplectic structure, and hence, it is a symplectic manifold. Recently, new invariants for symplectic manifolds in terms of cohomologies of differential forms have been introduced by Tseng and Yau. Here, we discuss the physical motivation behind the new symplectic invariants and analyze these invariants for phase space, i.e., the noncompact cotangent bundle.

Some applications of the LeeYang theorem
View Description Hide DescriptionFor lattice systems of statistical mechanics satisfying a LeeYang property (i.e., for which the LeeYang circle theorem holds), we present a simple proof of analyticity of (connected) correlations as functions of an external magnetic fieldh, for . A survey of models known to have the LeeYang property is given. We conclude by describing various applications of the aforementioned analyticity in h.

Simulating selfavoiding walks in bounded domains
View Description Hide DescriptionLet D be a domain in the plane containing the origin. We are interested in the ensemble of selfavoiding walks (SAWs) in D which start at the origin and end on the boundary of the domain. We introduce an ensemble of SAWs that we expect to have the same scaling limit. The advantage of our ensemble is that it can be simulated using the pivot algorithm. Our ensemble makes it possible to accurately study SchrammLoewner evolution (SLE) predictions for the SAW in bounded simply connected domains. One such prediction is the distribution along the boundary of the endpoint of the SAW. We use the pivot algorithm to simulate our ensemble and study this density. In particular the lattice effects in this density that persist in the scaling limit are seen to be given by a purely local function.
