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.     

Mean Field Limit for Bosons and Infinite Dimensional Phase-Space Analysis

This article proposes the construction of Wigner measures in the infinite dimensional bosonic quantum field theory, with applications to the derivation of the mean field dynamics. Once these asymptotic objects are well defined, it is shown how they can be used to make connections between different kinds of results or to prove new ones. Submitted: January 1, 2008., Accepted: July 1, 2008.     

Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data

The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non‐self‐adjoint scattering problem and ill‐posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0(x) = qχ[?L,L] for real amplitudes q. Using Riemann‐Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus‐one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data. © 2013 Wiley Periodicals, Inc.     

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011) to large volumes.     

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model that I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle’s position distribution converges to a Gaussian under diffusive rescaling.     

On the Semiclassical Limit of the General Modified NLS Equation

We study the semiclassical limit of the so-called general modified nonlinear Schrödinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity are proved for the dispersion limit of the cubic nonlinear case. The limiting transition from the MNLS equation to the NLS equation is also discussed.     

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

The Semiclassical Limit in the Quantum Drift-Diffusion Equations with Isentropic Pressure

The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.     

Mean-Field Limit and Semiclassical Expansion of a Quantum Particle System

We consider a quantum system constituted by N identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit N → ∞, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care of the dependence by considering the semiclassical expansion of the N-particle system. We prove that each term of the expansion agrees, in the limit N → ∞, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner formalism, which seems to be the most appropriate for the present problem. Submitted: October 2, 2008., Accepted: December 4, 2008.     

Correspondence between the NLS equation for optical fibers and a class of integrable NLS equations

The propagation of the optical field complex envelope in a single‐mode fiber is governed by a one‐dimensional cubic nonlinear Schrödinger equation with a loss term. We present a result about L²‐closeness of the solutions of the aforementioned equation and of a one‐dimensional nonlinear Schrödinger equation that is Painlevé integrable. Copyright © 2014 John Wiley & Sons, Ltd.     

The defocusing energy-supercritical NLS in higher dimensions

We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.     

The Existence of the Thermodynamic Limit for the System of Interacting Quantum Particles in Random Media

The thermodynamic limit of the internal energy and the entropy of the system of quantum interacting particles in random medium is shown to exist under the crucial requirements of stability and temperedness of interactions. The energy turns out to be proportional to the number of particles and/or volume of the system in the thermodynamic limit. The obtained results require very general assumptions on the random one-particle model. The methods are mainly based on subadditive type of inequalities.     

The NLS approximation for two dimensional deep gravity waves

Science China Mathematics - This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic...     

Quantum Analogue of Unstable Limit Cycles of a Periodically Perturbed Inverted Oscillator

Theoretical and Mathematical Physics - To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal...     

Inverse Problems for Obstacles in a Waveguide

In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ?X. In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry.     

Long time behavior for semiclassical NLS

In a recent paper, Jin, Levermore and McLaughlin analyze the semiclassical behavior of solutions to the defocusing, completely integrable nonlinear Schrödinger equation. We complete their analysis, by providing the long time behavior of the semiclassical solutions.     

Scattering for focusing combined power-type NLS

We consider the scattering of Cauchy problem for the focusing combined power-type Schr¨odinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi.