Derivation of the cubic non-linear Schrödinger 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 Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.     

Bose–Einstein Condensate and Singularities of the Frequency Dispersion of the Permittivity in a Disordered Coulomb System

In the framework of linear response theory, we consider the frequency dispersion of the permittivity of a disordered Coulomb system in the presence of the one-particle Bose–Einstein condensate for nuclei. We show that the superconductivity of nuclei exists in such a system and is manifested in the Meissner effect for a weakly nonuniform low-frequency electromagnetic field. The obtained result offers an opportunity to solve the problem of the presence of the one-particle Bose–Einstein condensate in superfluid He-II based on direct experiments.     

Thermodynamic Equilibrium in the System of Chaotic Quantized Vortices in a Weakly Imperfect Bose Gas

In the example of a weakly imperfect Bose gas, we discuss the mechanism of establishing thermodynamic equilibrium for a chaotic set of quantum vortex filaments. We assume that the dynamics of the Bose condensate is described by the Gross–Pitaevsky equation with an additional noise satisfying the fluctuation–dissipation theorem. In considering a vortex filament as the intersection line of surfaces on which the real and imaginary parts of the order parameter (x,t) vanish, we obtain an equation of the Langevin type for elements of the vortex filament with an appropriately transformed random force. The Fokker–Planck equation for the probability density has a solution given by the Gibbs distribution at the temperature of the Bose condensate. In other words, when the Bose condensate is in thermal equilibrium and no other random actions exist, the system of vortices is also in thermal equilibrium.     

Harmonic analysis on perturbed Cayley Trees

We study some spectral properties of the adjacency operator of non-homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. Apart from the natural mathematical meaning, such spectral properties are relevant for the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non-homogeneous networks. The resulting topological model is described by a one particle Hamiltonian which is, up to an additive constant, the opposite of the adjacency operator on the graph. It is known that the Bose Einstein condensation already occurs for unperturbed homogeneous Cayley Trees. However, the particles condensate on the perturbed graph, even in the configuration space due to non-homogeneity. Even if the graphs under consideration are exponentially growing, we show that it is enough to perturb in a negligible way the original graph in order to obtain a new network whose mathematical and physical properties dramatically change. Among the results proved in the present paper, we mention the following ones. The appearance of the Hidden Spectrum near the zero of the Hamiltonian, or equivalently below the norm of the adjacency. The latter is related to the value of the critical density and then with the appearance of the condensation phenomena. The investigation of the recurrence/transience character of the adjacency, which is connected to the possibility to construct locally normal states exhibiting the Bose Einstein condensation. Finally, the study of the volume growth of the wave function of the ground state of the Hamiltonian, which is nothing but the generalized Perron Frobenius eigenvector of the adjacency. This Perron Frobenius weight describes the spatial distribution of the condensate and its shape is connected with the possibility to construct locally normal states exhibiting the Bose Einstein condensation at a fixed density greater than the critical one.     

On the Spectrum of the One-Particle Density Matrix

Functional Analysis and Its Applications - The one-particle density matrix $$\gamma(x, y)$$ is one of the key objects in quantum-mechanical approximation schemes. The self-adjoint operator...     

Nonideal Bose Gases: Correlation Inequalities and Bose Condensation

We consider two simple model systems describing effective repulsion in a nonideal Bose gas. The interaction Hamiltonians in these systems can be analytically represented as functions of the occupation number operators for modes with nonzero momenta (p0). One of these models contains an interaction term corresponding to repulsion of bosons with the mode p=0 and ensuring the thermodynamic superstability of the system; the other model does not contain such a term. We use the Bogoliubov–Dirac–Ginibre approximation and the method of correlation inequalities to prove that a Bose condensate can exist in these model systems. Because of the character of interaction, the condensate can be formed in the superstable case for any values of the spatial dimensions, temperature, and positive chemical potentials.     

Stochastic Description of a Bose�CEinstein Condensate

In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose?CEinstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross?CPitaevskii scaling, which allows to give a theoretical proof of Bose?CEinstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random ??interaction-set?? with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.     

Smolukhowski problem for degenerate Bose gases

We construct a kinetic equation simulating the behavior of degenerate quantum Bose gases with the collision rate proportional to the molecule velocity. We obtain an analytic solution of the half-space boundary-value Smolukhowski problem of the temperature jump at the interface between the degenerate Bose gas and the condensed phase. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 498–511, June, 2008.     

Matrix models and parquet approximation

This paper examines the comparison of planar and planar parquet approximation in the zero-dimensional hermitian matrix models. We discuss how the parquet approximation reproduces the results of a planar approach to matrix model φ ⁴, multitrace model, two-matrix model and the Goldstone matrix model.     

Implementing Symmetries of a Free Bose Field via White Noise Operators

Linear symmetries of a free Bose field are exploited in the framework of Hida's white noise functionals triple. General symplectic automorphisms on the single particle space are implemented by generalized operators. The intertwining operators are constructed in a physically intuitive way, characterized analytically in terms of symbols, and factorized into three fundamental parts according to Wick ordering procedure. In particular, the classical Shale's theorem is rederived.     

Multi-Particle Correlation Functions in a Trapped Bose Gas

Multi-particle correlation functions at nonzero temperatures in a trapped Bose gas for D = 3, 2, 1 dimensions are considered. It is shown that, at relatively large distances, the multi-particle correlators are expressed in terms of one-particle ones. Bibliography: 13 titles.     

On positivity preservation for free quantum fields

Let A be a real Bose or Fermi one-particle operator with ∥ A ∥ ? I. Using Kaplansky's density theorem, a simple proof is given of the fact that Γ(A), the operator in Fock space induced by A, is positivity preserving in the relevant L²-space.     

Necessary condition for the existence of the S matrix outside the scope of perturbation theory

Using the Maslov—Shvedov complex-germ method due to Maslov—Shvedov, we obtain a necessary condition for the existence of the quantum-field S matrix outside the scope of perturbation theory in the leading order of semiclassical approximation. This condition consists in that the tangent symplectic transformation to the evolution operator of the nonlinear classical field equation is realized by a unitary transformation of Fock space. It follows from the results of the book of Maslov and Shvedov that this condition always holds.     

Inverse closed ultradifferential subalgebras

In previous work we have shown that classical approximation theory provides methods for the systematic construction of inverse-closed smooth subalgebras. Now we extend this work to treat inverse-closed subalgebras of ultradifferentiable elements. In particular, Carleman classes and Dales–Davie algebras are treated. As an application the result of Demko, Smith and Moss, and Jaffard on the inverse of a matrix with exponential decay is obtained within the framework of a general theory of smoothness.     

A model for an interacting quantum field

We present here a close nonlinear analog to the free quantum field of Bose statistics, in which the linear one-particle space is replaced by a nonlinear infinite-dimensional Hermitian symmetric space D, and the quantum field is constructed as a Hilbert space of holomorphic functions on D.     

Approximations by linear operators in spaces of fuzzy continuous functions

In this work, we further develop the Korovkin-type approximation theory by utilizing a fuzzy logic approach and principles of neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by the classical analysis. In the conventional setting, the Korovkin-type approximation theory is developed for continuous functions. Here we extend it to the space of fuzzy continuous functions, which contains a great diversity of functions that are not continuous. Furthermore, we give several applications, demonstrating that our new approximation results are stronger than the classical ones.     

Many-Particle and Semiclassical Limit Transitions for Nonrelativistic Bosons in a Quantized Electromagnetic Field

Using the method of the analytic germ, we obtain a system of equations for the amplitudes of one-particle phase densities of a system of several species of classical particles with electromagnetic interaction. The corresponding equations result from an extremely complicated limit transition in the theory of bosons interacting with a quantized electromagnetic field rather than in the classical equations for N particles in a magnetic field. This transition implies a double limit: first, the limit of large numbers of particles and photons and, second, the semiclassical limit. Moreover, in the first of these limits under some additional assumptions, we obtain the equations that are the steady-state conditions for an action functional considered in a recent paper by Faddeev and Niemi.     

Bose condensate in the two-dimensional case,the <Emphasis Type="Italic">λ</Emphasis>-point,and the Thiess-Landau two-fluid model

We discuss the relation between the Bose condensate and economic crisis problems, number theory, and clusterization __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 174–176, April, 2009.     

A good approximation of modulated amplitude waves in Bose–Einstein condensates

In this paper, we present a perturbation method that utilizes Hamiltonian perturbation theory and averaging to analyze spatio-temporal structures in Gross–Pitaevskii equations and thereby investigate the dynamics of modulated amplitude waves (MAWs) in quasi-one-dimensional Bose–Einstein condensates with mean-field interactions. A good approximation for MAWs is obtained. We also explore dynamics of BECs with the nonresonant external potentials and scatter lengths varying periodically in detail using Hamiltonian perturbation theory and numerical simulations.     

A linear system‐free Gaussian RBF method for the Gross‐Pitaevskii equation on unbounded domains

Gaussian radial basis function (RBF) interpolation methods are theoretically spectrally accurate. However, in applications this accuracy is seldom realized due to the necessity of solving a very poorly conditioned linear system to evaluate the methods. Recently, by using approximate cardinal functions and restricting the method to a uniformly spaced grid (or a smooth mapping thereof), it has been shown that the Gaussian RBF method can be formulated in a matrix free framework that does not involve solving a linear system [ 1 ]. In this work, we differentiate the linear system‐free Gaussian (LSFG) method and use it to solve partial differential equations on unbounded domains that have solutions that decay rapidly and that are negligible at the ends of the grid. As an application, we use the LSFG collocation method to numerically simulate Bose‐Einstein condensates. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 389–401, 2012