braid structure and raising lowering operator formalism in sutherland model we algebraically construct the fock space of the sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors for this purpose we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta the operators exchanging eigenvalues of two pseudomomenta have been known all the eigenstates are systematically produced by starting from the ground state and multiplying these operators by it

an almost sure large deviation principle for the hopfield model we prove a large deviation principle for the finite dimensional marginals of the gibbs distribution of the macroscopic overlap parameters in the hopfield model in the case where the number of random patterns m as a function of the system size n satisfies lim sup m n n 0 in this case the rate function is independent of the disorder for almost all realizations of the patterns

ward type identities for the two dimensional anderson model at weak disorder using the particular momentum conservation laws in dimension d 2 we rewrite the anderson model in terms of low momentum long range fields at the price of introducing electron loops the corresponding loops satisfy a ward type identity hence are much smaller than expected this fact should be useful for the study of the weak coupling model in the middle of the spectrum of the free hamiltonian

random matrix theory and the anderson model this paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal insulator anderson transition for the anderson model in dimension d 3 the possible criterions used to define such a transition are presented it is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model various simplified versions for which rigorous results have been obtained in the past are discussed it includes a free probability approach the wegner n orbital model and a class of models proposed by disertori pinson and spencer comm math phys 232 83 124 2002 at last a recent work by magnen rivasseau and the author markov process and related fields 9 261 278 2003 is summarized it gives a toy modeldescribing the lowest order approximation of anderson model and it is proved that for d 2 its density of states is given by the semicircle distribution a short discussion of its extension to d 3 follows

integrable spin chain with reflecting end a new integrable spin chain of the haldane shastry type is introduced it is interpreted as the inverse square interacting spin chain with a reflecting end the lattice sites of this model consist of the square roots of the zeros of the laguerre polynomial using the exchange operator formalism the integrals of motion for the model are explicitly constructed

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

connection between yangian symmetry and the quantum inverse scattering method the quantum nonlinear schrodinger model with two component fermions exhibits a yangian symmetry when considered on an infinite interval we construct the generators of the yangian using dunkl operators we show an equivalence between the monodromy matrix constructed from the dunkl operators and that of the quantum inverse scattering method under the yangian algebra the space of states with a fixed particle number forms a tensor product representation of fundamental representations

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 spinssi andp random patterns 1 2 n 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 p h i 1 n i si 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 is uniform and at most the two componentsh 1 andh 2 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 wherefcw 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 i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc 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 e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 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

rigorous mean field model for coherent potential approximation anderson model with free random variables a model of a randomly disordered system with site diagonal random energy fluctuations is introduced it is an extension of wegner sn orbital model to arbitrary eigenvalue distribution in the electronic level space the new feature is that the random energy values are not assumed to be independent at different sites but free freeness of random variables is an analog of the concept of independence for noncommuting random operators a possible realization is the ensemble of randomly rotated matrices at different lattice sites the one and two particle green functions of the proposed hamiltonian are calculated exactly the eigenstates are extended and the conductivity is novanishing everywhere inside the band the long range behavior and the zero frequency limit of the two particle green function are universal with respect to the eigenvalue distribution in the electronic level space the solutions solve the cpa equation for the one and two particle green function of the corresponding anderson model thus our multisite model is a rigorous mean field model for the single site cpa we show how the lloyd model is included in our model and treat various kinds of noises

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

the distribution of the partition function of the hopfield model with finite number of patterns we derive the leading term in the large n asymptotic expansion of the partition function of the hopfield model with finite number of patterns we show that this leading order term is deterministic in the high temperature region in the low temperature region and at the critical point it is random with the distribution governed by normal or iterated exponential distributions

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

eigenvalue statistics for lattice hamiltonian with off diagonal disorder this short note deals with a certain kind of lattice hamiltonian with off diagonal disorder based on the exponential decay of the fractional moment of the green function we are able to prove that the properly rescaled eigenvalues of the random hamiltonian are distributed as a poisson point process with intensity measure given by the density of states one of the key step in this proof is the minami type estimate as a crucial ingredient we also use the minami type estimate to study some important properties of the random hamiltonian such as multiplicity of the eigenvalues and quantitative estimate of the localization centers

fluctuation of the free energy in the hopfield model we prove that in the ergodic region t j2 1 r the deviation of the total free energy of the hopfield neural network converges in distribution asn to a shifted gaussian variable moreover the free energy per site converges in probability to lim 1 n ln n

an efficient technique for evaluating direct sequence spread spectrum multiple access communications a technique is presented for obtaining bounds on the average probability of error for direct sequence spread spectrum multiple access ds ssma communications the technique is of interest because it yields arbitrarily right bounds involves a small amount of computation avoids numerical integrations and applies to many types of detection as an illustration the technique is applied to binary ds ssma communications an additive white gaussian noise channel and a coherent correlation receiver it is assumed that all the signature sequences are deterministic each transmitter is assumed to have the same power although the approach can accommodate the case of transmitters with unequal powers expressions are given for the density functions of the random variables that model the multiple access interference these expressions are used to obtain arbitrarily tight upper and lower bounds on the average probability of error without making a gaussian approximation or performing numerical integrations to incorporate the effects of multiple access interference

limits of infinite interaction radius dimensionality and the number of components for random operators with off diagonal randomness we consider random one body operators that are analogs of the statistical mechanics hamiltonians with a varying interaction radiusr the dimensionality of spaced and the number of the field components orbitals n we prove that all the moments of the green functions for nonreal energies of these operators converge asr d n to the products of the average green functions just as in the mean field approximation of statistical mechanics we find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity which is nonzero on the whole support of the integrated density of states

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

integrable n particle hamiltonians with yangian or reflection algebra symmetry we use the dunkl operator approach to construct one dimensional integrable models describing n particles with internal degrees of freedom these models are described by a general hamiltonian belonging to the centre of the yangian or the reflection algebra which ensures that they admit the corresponding symmetry in particular the open problem of the symmetry is answered for the bn type sutherland model with spin and for a generalized bn type nonlinear schrodinger hamiltonian

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically