Regularity for Eigenfunctions of Schr?dinger Operators

We prove a regularity result in weighted Sobolev (or Babu?ka?CKondratiev) spaces for the eigenfunctions of certain Schr?dinger-type operators. Our results apply, in particular, to a non-relativistic Schr?dinger operator of an N-electron atom in the fixed nucleus approximation. More precisely, let ${\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}$ be the weighted Sobolev space obtained by blowing up the set of singular points of the potential ${V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}}$ , ${x \in \mathbb{R}^{3N}}$ , ${b_j, c_{ij} \in \mathbb{R}}$ . If ${u \in L^2(\mathbb{R}^{3N})}$ satisfies ${(-\Delta + V) u = \lambda u}$ in distribution sense, then ${u \in \mathcal{K}_{a}^{m}}$ for all ${m \in \mathbb{Z}_+}$ and all a ?? 0. Our result extends to the case when b _j and c _ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a?<?3/2.     

Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator

We extend the classical Schur–Weyl duality between representations of the groups ${SL(n, \mathbb{C})}$ and ${\mathfrak{S}_N}$ to the case of ${SL(n, \mathbb{C})}$ and the infinite symmetric group ${\mathfrak{S}_\mathbb{N}}$ . Our construction is based on a “dynamic,” or inductive, scheme of Schur–Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which has not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand–Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter–Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.     

Liouville-Type Theorems for the Forced Euler Equations and the Navier–Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in ${\mathbb {R}^N}$ . If we assume “single signedness condition” on the force, then we can show that a ${C^1 (\mathbb {R}^N)}$ solution (v, p) with ${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$ is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying ${v(x) \to 0}$ as ${|x| \to \infty}$ , the condition ${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$ , which is stronger than the important D-condition, ${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203–215, 2007), using the self-similar Euler equations directly.     

On the Quasi-linear Elliptic PDE {-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_k a_k \delta_{s_k}} in Physics and Geometry

It is shown that for each finite number N of Dirac measures ${\delta_{s_n}}$ supported at points ${s_n \in {\mathbb R}^3}$ with given amplitudes ${a_n \in {\mathbb R} \backslash\{0\}}$ there exists a unique real-valued function ${u \in C^{0, 1}({\mathbb R}^3)}$ , vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form ${-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_{n=1}^N a_n \delta_{s_n}}$ . Moreover, ${u \in C^{\omega}({\mathbb R}^3\backslash \{s_n\}_{n=1}^N)}$ . The result can be interpreted in at least two ways: (a) for any number N of point charges of arbitrary magnitude and sign at prescribed locations s _n in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as |s| ?? ??; (b) for any number N of integral mean curvatures assigned to locations ${s_n \in {\mathbb R}^3 \subset{\mathbb R}^{1, 3}}$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime ${{\mathbb R}^{1, 3}}$ , having lightcone singularities over the s _n but being smooth otherwise, and whose height function vanishes as |s| ?? ??. No struts between the point singularities ever occur.     

Low-regularity Schrödinger maps,II: global well-posedness in dimensions <Emphasis Type="Italic">d</Emphasis> ≥ 3

In dimensions d ≥ 3, we prove that the Schrödinger map initial-value problem

$ \left\{ \begin{array}{l} \partial_ts=s\times\Delta_x s\hbox{ on }\mathbb{R}^d\times\mathbb{R};\\ s(0)=s_0 \end{array} \right. $

is globally well-posed for small data s ₀ in the critical Besov spaces \({\dot{B}_Q^{d/2}(\mathbb{R}^d;\mathbb{S}^2)}\), \({Q\in\mathbb{S}^2}\).

     

Cosmic Censorship of Smooth Structures

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard ${\mathbb{R}^4}$ . Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold N and ${\mathbb{R}}$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to ${N\times \mathbb{R}}$ . Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on (3 + 1)-dimensional spacetimes.     

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between ‘weakly nonassociative’ algebras $\mathbb{A}$ (for which the associator ( $\mathbb{A},\mathbb{A}^2 ,\mathbb{A}$ ) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus $\mathbb{A}$ ′ of { $\mathbb{A}$ ) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with matrix algebra $\mathbb{A}$ ′, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators describing nonlocal interactions in ${L^2(\Omega; d^n x)}$ , where ${\Omega \subset \mathbb{R}^n}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ , are open sets with a compact, nonempty boundary ${\partial\Omega}$ satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in ${L^2(\Omega; d^{n} x)}$ to Fredholm perturbation determinants associated with operators in ${L^2(\partial\Omega; d^{n-1} \sigma)}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ . This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrödinger operator on the half-line ${(0,\infty)}$ , in the case of local interactions, to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.     

Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ ₀ liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity.     

Random Parking, Euclidean Functionals, and Rubber Elasticity

We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions S of subsets of ${\mathbb{R}^d}$ and of point sets that are (almost) subadditive in their first variable. Denoting by ξ the random parking measure in ${\mathbb{R}^d}$ , and by ξ ^R the random parking measure in the cube Q _R =  (?R, R) ^d , we show, under some natural assumptions on S, that there exists a constant ${\overline{S} \in \mathbb{R}}$ such that $$\lim_{R \to +\infty} \frac{S(Q_R, \xi)}{|Q_R|} \, = \, \lim_{R \to +\infty} \frac{S(Q_R, \xi^{R})}{|Q_R|} \, = \, \overline{S}$$ almost surely. If ${\zeta \mapsto S(Q_R, \zeta)}$ is the counting measure of ${\zeta}$ in Q _R , then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.     

Symmetries in Generalized Kähler Geometry

We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\mathbb{C}\mathbb{P}^{N}We define the notion of a moment map and reduction in both generalized complex geometry and generalized K?hler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on , Hirzebruch surfaces, the blow up of at arbitrarily many points, and other toric varieties, as well as complex Grassmannians.     

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Near-flat space limit of strings on AdS 4×ℂℙ3

The non-linear nature of string theory on non-trivial backgrounds, related to the AdS/CFT correspondence, force one to look for simplifications. Two such simplifications proved to be useful in studying string theory. These are the pp-wave limit, which describes point-like strings, and the so-called “near-flat space” limit which connects two different sectors of string theory—pp-wave and “giant magnons”. Recently another example of AdS/CFT duality emerged—AdS ₄/CFT ₃, which suggests duality between CS theory and superstring theory on . In this paper we study the “near-flat space” limit of strings on an background and discuss possible applications of the limiting theory. R.C. Rashkov is on leave from Department of Physics, Sofia University, Bulgaria.     

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

Let H = ?Δ + V, where V is a real valued potential on ${\mathbb {R}^2}$ satisfying ${\|V(x)|\lesssim \langle x \rangle^{-3-}}$ . We prove that if zero is a regular point of the spectrum of H = ?Δ + V, then $${\| w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\mathbb{R}^2)} \lesssim \frac{1}{|t|\log^2(|t|)} \| w f\|_{L^1(\mathbb{R}^2)},\,\,\,\,\,\,\,\, |t| \geq 2}$$ , with w(x) = (log(2 + |x|))². This decay rate was obtained by Murata in the setting of weighted L ² spaces with polynomially growing weights.     

Non-Uniform Dependence on Initial Data of Solutions to the Euler Equations of Hydrodynamics

We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H ^s (Ω) topology for any ${s \in \mathbb{R}}We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H ^s (Ω) topology for any s ? \mathbbR{s \in \mathbb{R}} if the domain Ω is the (flat) torus \mathbbTⁿ=\mathbbRⁿ/2p\mathbbZⁿ{\mathbb{T}^n=\mathbb{R}^n/2\pi\mathbb{Z}^n} and for any s > 0 if the domain is the whole space \mathbbRⁿ{\mathbb{R}^n}.     

Equivalent Forms of the Bessis–Moussa–Villani Conjecture

The BMV conjecture for traces, which states that ${\text{Tr}}\;{\text{exp}}\left( {A - \lambda B} \right)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $\lambda \mapsto {\text{Tr}}\;\left( {A + \lambda B} \right)^p$ has only non-negative coefficients for all $A,B \geqslant 0,p \in \mathbb{N}$ and (ii) $\lambda \mapsto {\text{Tr}}\;\left( {A + \lambda B} \right)^{ - p}$ is the Laplace transform of a positive measure for $A,B \geqslant 0,p > 0$ .     

The Percolation Transition in the Zero-Temperature Domany Model

We analyze a deterministic cellular automaton σ ^?=(σ ⁿ :n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{N}$ . The state space $\mathcal{S}_\mathbb{H} = \left\{ { - 1, + 1} \right\}^\mathbb{H}$ consists of assignments of ?1 or +1 to each site of $\mathbb{H}$ and the initial state $\sigma ^0 = \left\{ {\sigma _{^x }^0 } \right\}_{x \in \mathbb{H}}$ is chosen randomly with P(σ ⁰ _x=+1)=p∈[0,1]. The sites of $\mathbb{H}$ are partitioned in two sets $\mathcal{A}$ and $\mathcal{B}$ so that all the neighbors of a site x in $\mathcal{A}$ belong to $\mathcal{B}$ and vice versa, and the discrete time dynamics is such that the σ ^? _x 's with ${x \in \mathcal{A}}$ (respectively, $\mathcal{B}$ ) are updated simultaneously at odd (resp., even) times, making σ ^? _x agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p=1/2 in the percolation models defined by σ ⁿ , for all times n∈[1,∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν, and η of the dependent percolation models defined by σ ⁿ , n∈[1,∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).     

Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster

Let $Z_N$ be the number of self-avoiding paths of length $N$ starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on ${\mathbb Z} ^d$ with parameter $p>p_c({\mathbb Z} ^d)$ . The object of this paper is to study the connective constant of the dilute lattice $\limsup _{N\rightarrow \infty } Z_N^{1/N}$ , which is a non-random quantity. We want to investigate if the inequality $\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}$ obtained with the Borel–Cantelli Lemma is strict or not. In other words, we want to know if the quenched and annealed versions of the connective constant are equal. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when $d$ is sufficiently large there exists $p^{(2)}_c>p_c$ such that the inequality is strict for $p\in (p_c,p^{(2)}_c)$ .