A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.     

The ground state energy of a classical gas

In this paper we study the ground state energy of a classical gas. Our interest centers mainly on Coulomb systems. We obtain some new lower bounds for the energy of a Coulomb gas. As a corollary of our results we can show that a fermionic system with relativistic kinetic energy and Coulomb interaction is stable. More precisely, letH _N (α) be theN particle Hamiltonian $$H_N (\alpha ) = \alpha \sum\limits_{i = 1}^N {( - \Delta _i )^{1/2} + } \sum\limits_{i< j} {\left| {x_i - x_j } \right|^{ - 1} } - \sum\limits_{i,j} {\left| {x_i - R_j } \right|^{ - 1} } + \sum\limits_{i< j} {\left| {R_i - R_j } \right|^{ - 1} } $$ where Δ _i is the Laplacian in the variablex _i ∈?³ andR ₁, ...,R _N are fixed points in ?³. We show that for sufficiently large α, independent ofN, the HamiltonianH _N (α) is nonnegative on the space of square integrable functions ψ(x ₁, ...,x _N ), antisymmetric in the variablesx _i , 1≦i≦N.     

The ground state energy of a Bose gas with Coulomb interaction

LetH _N be the 2N particle Hamiltonian $$\begin{array}{*{20}c} {H_N = \sum\limits_{i = 1}^{2N} {( - \Delta _\iota ) + \sum\limits_{i< j = 1}^N {\left| {x_i - x_j } \right|^{ - 1} + } \sum\limits_{i< j = 1}^N {\left| {x_{i + N} - x_{j + N} } \right|^{ - 1} } } } \\ { - \sum\limits_{i,j< j = 1}^N {\left| {x_i - x_{j + N} } \right|^{ - 1} ,} } \\ \end{array} $$ whereΔ _i is the Laplacian in the variablex _i ∈?³, 1≦i≦2N. The operatorH _N is assumed to act on wave functionsΨ(x ₁, ...,x _N ;x _N+1, ...,x _2N ) which are symmetric in the variables (x ₁, ...,x _N ) and (x _N+1, ...,x _2N ). SupposeΨ is supported in a setΛ ^2N , whereΛ is a cube in ?³. It is shown that if a normalized wave functionΨ can be written as a product of two wave functions $$\psi (x_1 ,...,x_N ;x_{N + 1} ,...,x_{2N} ) = \psi _1 (x_2 ,...,x_N )\psi _2 (x_{N + 1} ,...,x_{2N} ),$$ and the density of particles inΛ is constant, then 〈Ψ|H _N |Ψ〉≧?CN ^7/5 for some universal constantC.     

Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in whichc _l (t) can represent the concentration ofl-particle clusters or droplets in (say) a condensing vapour at timet, are $$\begin{array}{*{20}c} {{{dc_l (t)} \mathord{\left/ {\vphantom {{dc_l (t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = J_{l - 1} (t) - J_l (t)} & {(l = 2,3,...)} \\ \end{array} $$ with $$J_l (t): = a_l c_1 (t)c_l (t) - b_{l + 1} c_{l + 1} (t)$$ and eitherc ₁=const. (‘case A’) or \(\rho : = \sum\limits_1^\infty {lc_l } \) =const. (‘case B’). The equilibrium solutions arec _l =Q _l z ^l , where \(Q_l : = \prod\limits_2^l {({{a_{r - 1} } \mathord{\left/ {\vphantom {{a_{r - 1} } {b_r }}} \right. \kern-0em} {b_r }})} \) . The density of the saturated vapour, defined as \(\rho _s : = \sum\limits_1^\infty {lQ_l z_s ^l } \) , wherez _s is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficientsa _l andb _l , there is a class of “metastable” solutions of the equations, withc ₁?z _s small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An “exponentially long time” means one that increases more rapidly than any negative power of the given value ofc ₁?z _s (or, in caseB,ρ?ρ _s ) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large.     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

Pure point spectrum for discrete almost periodic Schrödinger operators

The finite difference Schrödinger operator on ? ^m is considered $$Hu_j = \left( {\sum\limits_{v = 1}^m { D_v^2 } } \right)u_j + \frac{1}{\varepsilon }q_j u_j ,u \in \ell ^2 (\mathbb{Z}^m ),$$ where \(\sum\limits_{v = 1}^m { D_v^2 } \) is the difference Laplacian inm dimensions. For ? sufficiently small almost periodic potentialsq _j are constructed such that the operatorH has only pure point spectrum. The method is an inverse spectral procedure, which is a modification of the Kolmogorov-Arnol'd-Moser technique.     

On integrable systems close to the toda lattice

We study the generalized discrete self-trapping (DST) system formulated in terms of the u(n) Lie-Poisson algebra as well as its noncompact analog given on the gl(n) algebra. The Hamiltonian is a quadratic-linear function of the algebra generators where the quadratic part consists of the squared generators of the Cartan subalgebra only: $$H = \sum\limits_{i = 1}^n {\frac{{\gamma _i }}{2}A_{ii}^2 + } \sum\limits_{i,j = 1}^n {m_{ij} } A_{ij} $$ Two integrable cases are discovered: one for the u(n) case and the other for the gl(n) case. The correspondingL-operators (2 × 2 andn ×n) are found which give the Lax representation for these systems. The integrable model on the gl(n) algebra looks like the Toda lattice because in this case,m _ij=c _iδ_ij-1. The corresponding 2 × 2L-operator satisfies the Sklyanin algebra.     

Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans la théorie des champs de reggeons

In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory. The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator: $$\begin{gathered} H_{\lambda ',\mu ,\lambda ,\alpha } = \lambda '\sum\limits_{j = 1}^N {A_j^{ * 2} A_j^2 + \mu \sum\limits_{j = 1}^N {A_j^ * A_j + i\lambda \sum\limits_{j = 1}^N {A_j^ * (A_j + A_j^ * )A_j } } } \hfill \\ + \alpha \sum\limits_{j = 1}^{N - 1} {(A_{j + 1}^ * A_j + A_j^ * A_{j + 1} ),} \hfill \\ \end{gathered}$$ whereA* _j andA _j ,j∈[1,N] are the creation and annihilation operators. In the third section, we restrict our study to the case of nul transverse dimension (N=1). Following the study done in [1], we consider the operator: $$H_{\lambda ',\mu ,\lambda } = \lambda 'A^{ * 2} A^2 + \mu A^ * A + i\lambda A^ * (A + A^ * )A,$$ whereA* andA are the creation and annihilation operators. For λ′>0 and λ′²≦μλ′+λ². We prove that the solutions of the equationu′(t)+H _{λ′, μ,λ} u(t)=0 are expandable in series of the eigenvectors ofH _λ′,μ,λ fort>0. In the last section, we show that the smallest eigenvalue σ(α) of the operatorH _{λ′,μ,λ,α} is analytic in α, and thus admits an expansion: σ(α)=σ₀+ασ₁+α²σ₂+..., where σ₀ is the smallest eigenvalue of the operatorH _{λ′,μ,λ,0}.     

One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJ _ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0} $$ , ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ^∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 _H denotes the Gibbs average with respect to the HamiltonianH.     

On a third-order phase transition

The asymptotic behaviour of random variables of the general form $$\ln \sum\limits_{i = 1}^{\kappa ^N } {\exp (N^{1/p} \beta \zeta _i )} $$ with independent identically distributed random variables ζ _i is studied. This generalizes the random energy model of Derrida. In the limitN→∞, there occurs a particular kind of phase transition, which does not incorporate a bifurcation phenomenon or symmetry breaking. The hypergeometric character of the problem (see definitions of Sect. 4), its Φ-function, and its entropy function are discussed.     

Absence of long-range order in one-dimensional spin systems

For a one-dimensional Ising model with interaction energy $$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i< j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$ it is proved that there is no long-range order at any temperature when $$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$      

Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i < j \leqslant N} } V_{ij} \left( {x_i - x_j } \right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.     

Supersymmetric particle model with additional bosonic coordinates

A new supersymmetric particle model in enlarged superspace with additional bosonic coordinatesz _ij , \(\bar z_{ij} \) (z _ij =?z _ji ;i=1...N, N even) canonically conjugated to central charges is quantized. The superwave functions which are obtained through first quantization are the free superfields on the enlarged superspace \((x^\mu , \theta _{\alpha i} , \bar \theta _i^{\dot \alpha } , z_{ij} , \bar z_{ij} )\) . Two particular cases (N=2 with one additional complex bosonic coordinate andN=8 with seven additional real coordinates) are considered in more detail.     

Exponential lower bounds to solutions of the Schrödinger equation: Lower bounds for the spherical average

For a large class of generalizedN-body-Schrödinger operators,H, we show that ifE<Σ=infσ_ess(H) and ψ is an eigenfunction ofH with eigenvalueE, then $$\begin{array}{*{20}c} {\lim } \\ {R \to \infty } \\ \end{array} R^{ - 1} \ln \left( {\int\limits_{S^{n - 1} } {|\psi (R\omega )|} ^2 d\omega } \right)^{1/2} = - \alpha _0 ,$$ with α ₀ ² +E a threshold. Similar results are given forE≧Σ.     

Infinite differentiability for one-dimensional spin system with long range random interaction

We consider one-dimensional spin systems with Hamiltonian: $$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$ , where ? _tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α>3/2 andE(? _tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ^∞ both inh and β. Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ? _tt′ 's, where the quenched magnetization isC ^∞ but fails to be analytic inh for suitableh and β.     

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature \(\widetilde\beta \) and prove that, in the limitN→∞, \(\widetilde\beta \) /N→β, αN→1, where β∈(?8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ?^N = ? ^N x,x∈Λ converges to a superposition of solutions ? _α of the following Mean Field Equation: $$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$ Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→?8π⁺, either ?_β → δ _{x 0} (weakly in the sense of measures) wherex ₀ denotes and equilibrium point of a single point vortex in Λ, or ?_β converges to a smooth solution of (A.1) for β=?8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.     

Dobrushin states for classical spin systems with complex interactions

We consider a classical spin system on the hypercubic lattice with a general interaction of the form $$ H = \frac{\beta } {4}\sum\limits_{\begin{array}{*{20}c} {x,y:} \\ {|x - y| = 1} \\ \end{array} } {|s_x - s_y | - h} \sum\limits_x {x{}_x + } \sum\limits_A {\lambda _A \prod\limits_{y \in A} {S_y } } $$ are the spin variables, Β is the inverse temperature,h is the magnetic field, andλ _A are translation-invariant coupling constants satisfyingλ _A = 0 if diamA > l. No symmetry relating the configurationss ={sinx} and-s=-s _x is assumed. In dimension d-3, we construct low-temperature States which break the translation invariance of the system by introducing so-called Dobrushin boundary conditions which force a horizontal interface into the system. In contrast to previous constructions, our methods work equally well for complex interactions, and should therefore be generalizable to quantum spin systems.