on the mean field and classical limits of quantum mechanics the main result in this paper is a new inequality bearing on solutions of the n body linear schrodinger equation and of the mean field hartree equation this inequality implies that the mean field limit of the quantum mechanics of n identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation this result applies to the case of c1 1 interaction potentials the quantity measuring the approximation of the n body quantum dynamics by its mean field limit is analogous to the monge kantorovich or wasserstein distance with exponent 2 the inequality satisfied by this quantity is reminiscent of the work of dobrushin on the mean field limit in classical mechanics func anal appl 13 115 123 1979 our approach to this problem is based on a direct analysis of the n particle liouville equation and avoids using techniques based on the bbgky hierarchy or on second quantization

absolute value equations abstract we investigate existence and nonexistence of solutions for np hard equations involving absolute values of variables ax x b where a is an arbitrary n n real matrix by utilizing an equivalence relation to the linear complementarity problem lcp we give existence results for this class of absolute value equations aves as well as a method of solution for special cases we also give nonexistence results for our ave using theorems of the alternative and other arguments

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

derivation of the gross pitaevskii hierarchy for the dynamics of bose einstein condensate consider a system of n bosons on the three dimensional unit torus interacting via a pair potential n2v n xi xj where x x1 xn denotes the positions of the particles suppose that the initial data n 0 satisfies the condition where hn is the hamiltonian of the bose system this condition is satisfied if n 0 wn n 0 where wn is an approximate ground state to hn and n 0 is regular let n t denote the solution to the schrodinger equation with hamiltonian hn gross and pitaevskii proposed to model the dynamics of such a system by a nonlinear schrodinger equation the gross pitaevskii gp equation the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices k ut ut solves the gp hierarchy we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy our analysis requires that the n boson dynamics be described by a modified hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance our proof can be extended to a modified hamiltonian that only forbids at least n particles from coming close together for any fixed n 2006 wiley periodicals inc

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

on well posedness for general hierarchy equations of gross pitaevskii and hartree type gross pitaevskii and hartree hierarchies are infinite systems of coupled pdes emerging naturally from the mean field theory of bose gases their solutions are known to be related to initial value problems in particular the gross pitaevskii and hartree equations due to their physical and mathematical relevance the issues of well posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques in this article we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain liouville equations and secondly solving the uniqueness and existence questions for the latter as an outcome we formulate a hierarchy equation starting from any initial value problem which is u 1 invariant and prove a general principle which can be stated formally as follows in particular several new well posedness results as well as a counterexample to uniqueness for the gross pitaevskii hierarchy equation are proved the novelty in our work lies in the aforementioned duality and the use of liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

a dynamic model to solve the absolute value equations abstract in this paper the analytic solution of the absolute value equations ave is investigated as far as we know there are some numerical methods to obtain the solution of the ave however there is not a study on exact solution of the ave here we try to obtain the exact solution of the ave based on a dynamical system model constructed by the projection function finally the simulation results show the effectiveness and the accuracy of the method

on the uniqueness of solutions to the periodic 3d gross pitaevskii hierarchy abstract in this paper we present a uniqueness result for solutions to the gross pitaevskii hierarchy on the three dimensional torus under the assumption of an a priori spacetime bound we show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear schrodinger equation in this way we obtain a periodic analogue of the uniqueness result on r 3 previously proved by klainerman and machedon 75 except that in the periodic setting we need to assume additional regularity in particular we need to work in the sobolev class h for 1 by constructing a specific counterexample we show that on t 3 the existing techniques from the work of klainerman and machedon approach do not apply in the endpoint case 1 this is in contrast to the known results in the non periodic setting where these techniques are known to hold for all 1 in our analysis we give a detailed study of the crucial spacetime estimate associated to the free evolution operator in this step of the proof our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice this method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius r depending on the number theoretic properties of the normal vector to the plane we are hence able to obtain a sharp range of admissible sobolev exponents for which the spacetime estimate holds

the quintic nls as the mean field limit of a boson gas with three body interactions we investigate the dynamics of a boson gas with three body interactions in dimensions d 1 2 we prove that in the limit of infinite particle number the bbgky hierarchy of k particle marginals converges to a limiting gross pitaevskii gp hierarchy for which we prove existence and uniqueness of solutions factorized solutions of the gp hierarchy are shown to be determined by solutions of a quintic nonlinear schrodinger equation our proof is based on and extends methods of erdos schlein yau klainerman machedon and kirkpatrick schlein staffilani

dimension reduction of the schrodinger equation with coulomb and anisotropic confining potentials we consider dimension reduction for the three dimensional 3d schrodinger equation with the coulomb interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model sam or surface density model sdm in two dimensions 2d and the line adiabatic model lam in one dimension 1d efficient and accurate numerical methods for computing ground states and dynamics of the sam sdm and lam models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models they are applied to find numerically convergence and convergence rates for the dimension reduction from 3d to 2d and 3d to 1d in terms of ground state and dynamics which confirm some existing analytical results for the dimension reduction in the literature in particular we explain and demonstrate that the standard schrodinger poisson system in 2

randomization and the gross pitaevskii hierarchy we study the gross pitaevskii hierarchy on the spatial domain mathbb t 3 by using an appropriate randomization of the fourier coefficients in the collision operator we prove an averaged form of the main estimate which is used in order to contract the duhamel terms that occur in the study of the hierarchy in the averaged estimate we do not need to integrate in the time variable an averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary the range of regularity exponents that we obtain is alpha frac 3 4 it was shown in our previous joint work with gressman j funct anal 266 7 4705 4764 2014 that the range alpha 1 is sharp in the corresponding deterministic spacetime estimate this is in contrast to the non periodic setting which was studied by klainerman and machedon commun math phys 279 1 169 185 2008 where the spacetime estimate is known to hold whenever alpha geq 1 the goal of our paper is to extend the range of in this class of estimates in a probabilistic sense we use the new estimate and the ideas from its proof in order to study randomized forms of the gross pitaevskii hierarchy more precisely we consider hierarchies similar to the gross pitaevskii hierarchy but in which the collision operator has been randomized for these hierarchies we show convergence to zero in low regularity sobolev spaces of duhamel expansions of fixed deterministic density matrices we believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

on a system of equations for the normal fluid condensate interaction in a bose gas abstract the existence of global solutions for a system of differential equations is proved and some of their properties are described the system involves a kinetic equation for quantum particles it is a simplified version of a mathematical description of a weakly interacting dilute gas of bosons in the presence of a condensate near the critical temperature

commuting finite blaschke products with no fixed points in the unit disk abstract in this paper we study when two finite blaschke products commute we complete previous results by chalendar and mortini when they have a fixed point in the unit disk and by arteaga when they do not have a fixed point in the unit disk

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

negative energy ground states for the l2 critical nlse on metric graphs we investigate the existence of ground states with prescribed mass for the focusing nonlinear schrodinger equation with l2 critical power nonlinearity on noncompact quantum graphs we prove that unlike the case of the real line for certain classes of graphs there exist ground states with negative energy for a whole interval of masses a key role is played by a thorough analysis of gagliardo nirenberg inequalities and by estimates of the optimal constants most of the techniques are new and suited to the investigation of variational problems on metric graphs

sor like iteration method for solving absolute value equations in this paper we propose an sor like iteration method for solving the absolute value equation ave which is obtained by reformulating equivalently the ave as a two by two block nonlinear equation the convergence results of the proposed iteration method are proved under certain assumptions imposed on the involved parameter numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the sor like iteration method

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

solving absolute value equation using complementarity and smoothing functions in this paper we reformulate the np hard problem of the absolute value equation ave as a horizontal linear complementarity one and then solve it using a smoothing technique this approach leads to a new class of methods that are valid for general absolute value equation an asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates this kind of error bound or estimate had never been studied for other known methods the corresponding algorithms were tested on randomly generated problems and applications these experiments show that in the general case one observes a reduction of the number of failures

asymptotic behavior of the nonlinear schrodinger equation with harmonic trapping we consider the cubic nonlinear schrodinger equation with harmonic trapping on d 1 d 5 in the case when all directions but one are trapped aka cigar shaped trap we prove modified scattering and construct modified wave operators for small initial and final data respectively the asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the nls equation with full trapping on d 1 in the physical dimension d 3 this system turns out to be exactly the cr equation derived by faou germain and the first author as the large box limit of the resonant nls equation in the homogeneous zero potential setting the special dynamics of the latter equation combined with the above modified scattering results allow us to justify and extend some physical approximations in the theory of bose einstein condensates in cigar shaped traps 2015 wiley periodicals inc

on the uniqueness of solutions to the gross pitaevskii hierarchy the purpose of this note is to give a new proof of uniqueness of the gross pitaevskii hierarchy first established in 1 in a different space based on space time estimates similar in spirit to those of 2

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

fixed points of commuting holomorphic mappings other than the wolff point let be the unit disc of c and let f g hol be such that f o g g o f for a 1 let fix a f p l lim r 1 f rp p lim r 1 f rp a we study the behavior of g on fix a f in particular we prove that g fix a f fix a f as a consequence besides conditions for fix a f fix a g 0 we prove a conjecture of c cowen in case f and g are univalent mappings

strichartz estimates for schrodinger equations on irrational tori in this paper we prove new strichartz estimates for linear schrodinger equations posed on d dimensional irrational tori then we use these estimates to prove subcritical and critical local well posedness results for nonlinear schrodinger equations on irrational tori

rigorous derivation of the cubic nls in dimension one we derive rigorously the one dimensional cubic nonlinear schrodinger equation from a many body quantum dynamics the interaction potential is rescaled through a weak coupling limit together with a short range one we start from a factorized initial state and prove propagation of chaos with the usual two step procedure in the former step convergence of the solution of the bbgky hierarchy associated to the many body quantum system to a solution of the bbgky hierarchy obtained from the cubic nls by factorization is proven in the latter we show the uniqueness for the solution of the infinite bbgky hierarchy

on the cauchy problem for gross pitaevskii hierarchies the purpose of this paper is to investigate the cauchy problem for the gross pitaevskii infinite linear hierarchy of equations on rn n 1 we prove local existence and uniqueness of solutions in certain sobolev type spaces h of sequences of marginal density operators with n 2 in particular we give a clear discussion of all cases n 2 which covers the local well posedness problem for the gross pitaevskii hierarchy in this situation

a simple proof of convergence to the hartree dynamics in sobolev trace norms the derivation of the hartree equation from many body systems of bosons in the mean field limit has been very intensively studied in the last couple of years however very few results exist showing convergence of the kth marginal of the n body density matrix to the projection to the k fold tensor product of the solution of the hartree equation in stronger trace norms like the energy trace norm see the work of michelangeli and schlein commun math phys 311 3 645 687 2012 and luhrmann j math phys 53 2 022105 2012 this issue is from a physical view point very important the reason is that one can then approximate expectation values of certain observables of the n body system by means of the hartree equation with relaxation of the very restrictive assumption that the observables are bounded operators here we consider the non relativistic case we prove assuming only h1 regularity of the initial data convergence in the energy trace norm without rates and convergence in any other weaker

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

uniqueness of solutions to the 3d quintic gross pitaevskii hierarchy abstract in this paper we study solutions to the three dimensional quintic gross pitaevskii hierarchy we prove unconditional uniqueness among all small solutions in the critical space h 1 which corresponds to h 1 on the nls level with slight modifications to the proof we also prove unconditional uniqueness of solutions to the hartree hierarchy without a smallness condition our proof uses the quantum de finetti theorem and is an extension of the work by chen hainzl pavlovic and seiringer 9 and our previous work 32

the mean field approximation and the non linear schrodinger functional for trapped bose gases we study the ground state of a trapped bose gas starting from the full many body schrodinger hamiltonian and derive the nonlinear schrodinger energy functional in the limit of large particle number when the interaction potential converges slowly to a dirac delta function our method is based on quantitative estimates on the discrepancy between the full many body energy and its mean field approximation using hartree states these are proved using finite dimensional localization and a quantitative version of the quantum de finetti theorem our approach covers the case of attractive interactions in the regime of stability in particular our main new result is a derivation of the 2d attractive nonlinear schrodinger ground state

smooth feshbach map and operator theoretic renormalization group methods abstract a new variant of the isospectral feshbach map defined on operators in hilbert space is presented it is constructed with the help of a smooth partition of unity instead of projections and is therefore called smooth feshbach map it is an effective tool in spectral and singular perturbation theory as an illustration of its power a novel operator theoretic renormalization group method is described and applied to analyze a general class of hamiltonians on fock space the main advantage of the new renormalization group method over its predecessors is its technical simplicity which it owes to the use of the smooth feshbach map

on the mean field approximation of many boson dynamics abstract we show under general assumptions that the mean field approximation for quantum many boson systems is accurate our contribution unifies and improves most of the known results the proof uses general properties of quantization in infinite dimensional spaces phase space analysis and measure transportation techniques

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

a central limit theorem in many body quantum dynamics 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

on 2d nonlinear schrodinger equations with data on r t abstract we prove l 2 global well posedness results for 2d subcritical and critical nonlinear schrodinger equations with data on r t we use methods of the periodic case due to j bourgain the main ingredient in the proof is the l 4 l 2 strichartz inequality for the free evolution which fails in the purely periodic setting

rigorous derivation of the gross pitaevskii equation with a large interaction potential consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 hn n 0i cn and that the one particle density matrix converges to a projection as n then we prove that the k particle density matrices of n t factorize in the limit n moreover the one particle orbital wave function solves the time dependent gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant proportional to the scattering length of the potential v in 12 we proved the same statement under the condition that the interaction potential v is sufficiently small in the present work we develop a new approach that requires no restriction on the size of the potential

the derivation of the mathbb t 3 t 3 energy critical nls from quantum many body dynamics we derive the 3d energy critical quintic nls from quantum many body dynamics with 3 body interaction in the mathbb t 3 periodic setting due to the known complexity of the energy critical setting previous progress was limited in comparison to the 2 body interaction case yielding energy subcritical cubic nls previously the only result for the 3d energy critical case was hong et al j funct anal 270 1 34 67 2016 which proved the uniqueness part of the argument in the case of small solutions in the main part of this paper we develop methods to prove the convergence of the bbgky hierarchy to the infinite gross pitaevskii gp hierarchy and separately the uniqueness of large gp solutions since the trace estimate used in the previous proofs of convergence is the false endpoint trace estimate in our setting we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de finetti theorem for the large solution uniqueness argument we discover the new hierarchical uniform frequency localization hufl property for the gp hierarchy and use it to prove a new type of uniqueness theorem the hufl property reduces to a new statement even for nls with the help of colliander et al ann math 2 167 3 767 865 2008 and ionescu and pausader duke math j 161 1581 1612 2012 which proved the global well posedness for the quintic nls this new uniqueness theorem establishes global uniqueness

derivation of the time dependent gross pitaevskii equation without positivity condition on the interaction using a new method pickl in a simple derivation of mean field limits for quantum systems 2010 it is possible to derive mean field equations from the microscopic n body schrodinger evolution of interacting particles without using bbgky hierarchies in this paper we wish to analyze scalings which lead to the gross pitaevskii equation which is usually derived assuming positivity of the interaction erdos et al in commun pure appl math 59 12 1659 1741 2006 invent math 167 515 614 2007 the new method for dealing with mean field limits presented in pickl 2010 allows us to relax this condition the price we have to pay for this relaxation is however that we have to restrict the scaling behavior of the interaction and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function mu psi 0 to a pure state 0 0

quantitative derivation of the gross pitaevskii equation starting from first principle many body quantum dynamics we show that the dynamics of bose einstein condensates can be approximated by the time dependent nonlinear gross pitaevskii equation giving a bound on the rate of the convergence initial data are constructed on the bosonic fock space applying an appropriate bogoliubov transformation on a coherent state with expected number of particles mathit n the bogoliubov transformation plays a crucial role it produces the correct microscopic correlations among the particles our analysis shows that on the level of the one particle reduced density the form of the initial data is preserved by the many body evolution up to a small error that vanishes as mathit n 1 2 in the limit of large mathit n

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

global well posedness of the energy critical nonlinear schrodinger equation with small initial data in h1 t3 a refined trilinear strichartz estimate for solutions to the schrodinger equation on the flat rational torus t3 is derived by a suitable modification of critical function space theory this is applied to prove a small data global well posedness result for the quintic nonlinear schrodinger equation in hs t3 for all s 1 this is the first energy critical global well posedness result in the setting of compact manifolds

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

the stable subset of a univalent self map we give a complete description of the stable subset the union of all backward orbit with bounded step and of the pre models of a univalent self map f x rightarrow x where x is a kobayashi hyperbolic cocompact complex manifold such as the ball or the polydisc in mathbb c q the result is obtained studying the complex structure of a decreasing intersection of complex manifolds all biholomorphic to x

mean field evolution of fermionic mixed states in this paper we study the dynamics of fermionic mixed states in the mean field regime we consider initial states that are close to quasi free states and prove that under suitable assumptions on the initial data and on the many body interaction the quantum evolution of such initial data is well approximated by a suitable quasi free state in particular we prove that the evolution of the reduced one particle density matrix converges as the number of particles goes to infinity to the solution of the time dependent hartree fock equation our result holds for all times and gives effective estimates on the rate of convergence of the many body dynamics towards the hartree fock evolution 2015 wiley periodicals inc

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

the monotone convergence rate of the parallel nonlinear aor method abstract a new comparison theorem about the parallel nonlinear aor method 1 is set up which describes in detail the influence of either the multisplitting of the coefficient matrix or the pair of the relaxation parameters on the convergence rate of this method

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

the endpoint case of the bennett carbery tao multilinear kakeya conjecture we prove the endpoint case of the multilinear kakeya conjecture of bennett carbery and tao the proof uses the polynomial method introduced by dvir

mean field dynamics of a quantum tracer particle interacting with a boson gas abstract we consider the dynamics of a heavy quantum tracer particle coupled to a non relativistic boson field in r 3 the pair interactions of the bosons are of mean field type with coupling strength proportional to 1 n where n is the expected particle number assuming that the mass of the tracer particle is proportional to n we derive generalized hartree equations in the limit n moreover we prove the global well posedness of the associated cauchy problem for sufficiently weak interaction potentials

asymptotic stability of ground states in 2d nonlinear schrodinger equation including subcritical cases abstract we consider a class of nonlinear schrodinger equations in two space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l 2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a carefully chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

existence of dynamics for a 1d nls equation perturbed with a generalized point defect in the present paper we study the well posedness for the one dimensional cubic nls perturbed by a generic point interaction point interactions are described as the 4 parameter family of self adjoint extensions of the symmetric 1d laplacian defined on the regular functions vanishing at a point and in the present context can be interpreted as localized defects interacting with the nls field a previously treated special case is given by an nls equation with a defect which we generalize and extend as far as well posedness is concerned to the whole family of point interactions we prove existence and uniqueness of the local cauchy problem in strong form initial data and evolution in the operator domain of point interactions weak form initial data and evolution in the form domain of point interactions and conservation laws of mass and energy are proved for finite energy weak solutions of the problem which imply global existence of the dynamics a technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions to overcome it a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow

the kinetic limit of a system of coagulating brownian particles we consider a random model of diffusion and coagulation a large number of small particles is randomly scattered in mathbb r d at an initial time each particle has some integer mass and moves as a brownian motion whose rate of diffusion is determined by that mass when any two particles are close they are liable to combine into a single particle that bears the mass of each of them the range of interaction between pairs of particles is chosen so that a typical particle is liable to interact with a unit order of other particles in a unit of time we determine the macroscopic evolution of the system in any dimension d 3 the density of particles evolves according to the smoluchowski system of partial differential equations indexed by the mass parameter in which the interaction term is a sum of products of densities central to the proof is the task of establishing the so called stosszahlansatz which asserts that at any given time the presence of particles of two given masses at any given point in macroscopic space is asymptotically independent as the initial number of particles is taken to be high nonetheless there is in a microscopic region about each particle a reduced probability of finding another particle determining this deficit precisely is necessary in computing the coefficients appearing in the interaction terms of the smoluchowski partial differential equation

mean field limit for bosons with compact kernels interactions by wigner measures transportation we consider a class of many body hamiltonians composed of a free kinetic part and a multi particle potential interaction with a compactness assumption on the latter part we investigate the mean field limit of such quantum systems following the wigner measures approach we prove the propagation of these measures along the flow of a nonlinear hartree field equation this enhances and complements some previous results in the subject

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

quantum dynamics with mean field interactions a new approach we propose a new approach for the study of the time evolution of a factorized n particle bosonic wave function with respect to a mean field dynamics with a bounded interaction potential the new technique which is based on the control of the growth of the correlations among the particles leads to quantitative bounds on the difference between the many particle schrodinger dynamics and the one particle nonlinear hartree dynamics in particular the one particle density matrix associated with the solution to the n particle schrodinger equation is shown to converge to the projection onto the one dimensional subspace spanned by the solution to the hartree equation with a speed of convergence of order 1 n for all fixed times

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem abstract the linear complementarity problem lcp has many applications as e g in the solution of linear and convex quadratic programming in free boundary value problems of fluid mechanics etc in the present work we assume that the matrix coefficient m r n n ccccc cof the lcp is symmetric positive definite and we introduce the optimal nonstationary extrapolation to improve the convergence rates of the well known modulus algorithm and block modulus algorithm for its solution two illustrative numerical examples show that the optimal nonstationary extrapolated block modulus algorithm is far better than all the previous similar algorithms

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

commuting analytic functions let f and g not conformal automorphisms of the unit disk be analytic mappings of the unit disk into itself we say f and g commute if f o g g o f this paper characterizes those functions g that commute with a given function f several corollaries of this characterization give qualitative information about g given similar information about f and examples are given in each case to show the limitations of the conclusions some of the qualitative properties considered are univalence fixed point sets and whether two such g must commute with each other

the gross pitaevskii hierarchy on general rectangular tori in this work we study the gross pitaevskii hierarchy on general rational and irrational rectangular tori of dimensions two and three this is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear schrodinger equation we prove a conditional uniqueness result for the hierarchy in two dimensions this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear schrodinger equation from the dynamics of many body quantum systems on irrational tori this question was posed as an open problem in the previous work of kirkpatrick schlein and staffilani

norm approximation for many body quantum dynamics focusing case in low dimensions abstract we study the norm approximation to the schrodinger dynamics of n bosons in r d d 1 2 with an interaction potential of the form n d 1 w n x y here we are interested in the focusing case w 0 assuming that there is complete bose einstein condensation in the initial state we show that in the large n limit the evolution of the condensate is effectively described by a nonlinear schrodinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic hamiltonian resulting from bogoliubov approximation our result holds true for all 0 when d 1 and for all 0 1 when d 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

a justification of the modulation approximation to the 3d full water wave problem 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 schrodinger 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

construction of a gibbs measure associated to the periodic benjamin ono equation we define a finite borel measure of gibbs type supported by the sobolev spaces of negative indexes on the circle the measure can be seen as a limit of finite dimensional measures these finite dimensional measures are invariant by the ode s which correspond to the projection of the benjamin ono equation posed on the circle on the first n n 1 modes in the trigonometric bases

gross pitaevskii equation as the mean field limit of weakly coupled bosons we consider the dynamics of n boson systems interacting through a pair potential n 1va xi xj where va x a 3v x a we denote the solution to the n particle schrodinger equation by n t recall that the gross pitaevskii gp equation is a nonlinear schrodinger equation and the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices open image in new window solves the gp hierarchy under the assumption that a n for 0 3 5 we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy with the coupling constant in the nonlinear term of the gp equation given by v x dx the uniqueness of the solutions of this hierarchy remains an open question

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

parallel nonlinear aor method and its convergence abstract we set up a class of parallel nonlinear aor method in the sense of matrix multi splitting for solving the large scale system of nonlinear equations ax x b with a l rn nonsingular b rn and rn rn being continuously diagonal the global as well as the monotone convergence of this method is proved