A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations

A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0202309)This research was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0207064).     

On Asymptotic Nonlocal Symmetry of Nonlinear Schrödinger Equations

Abstract

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schrödinger equations. An important class of solutions of the free Schrödinger equation with improved smoothing properties is obtained.     

Asymptotic Properties of Solutions to 3-Particle Schrödinger Equations

We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ ^a V ^a (x ^a ) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ^* as the image of ℱ(λ)^*. These stationary solutions admit asymptotic expansions in ℬ^* in terms of spherical waves associated with scattering channels. Received: 20 September 2000 / Accepted: 20 May 2001     

Gauge Transformations for a Family of Nonlinear Schrödinger Equations

Abstract

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled nonlinear “Schrödinger-Maxwell” theory. Time-dependent gauge parameters result in additional terms of the type introduced by Kostin and Bialynicki-Birula and Mycielski, but Maxwell’s equations for the (non-quantized) gauge-invariant electric and magnetic fields remain linear.     

Dispersion of Singularities of Solutions for Schrödinger Equations

We consider the singularities of solutions for the Schrödinger evolution equation associated with where Q is a d×d real symmetric matrix with the eigenvalues ₁,,_d, and WC(R^d,R) satisfies W(x)=o(|x|²) as |x|. Under additional conditions, we show the dispersion of microlocal singularities of solutions due to the principal symbol in all directions at time and in the nondegenerate directions at t. We also show the weaker dispersion of microlocal singularities of solutions due to the subprincipal symbol W in the degenerate directions at t if W satisfies W(x)=O(|x|¹⁺) as |x| for some 0<<1 and additional conditions. In particular, we prove the dispersion of microlocal singularities of solutions at resonant times when H is a perturbed harmonic oscillator.Partly supported by Grand-in-Aid for Young Scientists (B) 14740110, Japan Society of the Promotion of Science; and Mathematical Sciences Research Institute in BerkeleyDedicated to Professor Mitsuru Ikawa on his sixtieth birthday     

On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

We consider nonlinear Schrödinger equations $iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,$ where d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called the Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I.M.Sigal.     

Symmetry Classification for a Coupled Nonlinear Schrödinger Equations

Abstract

We do a Lie symmetry classification for a system of two nonlinear coupled Schrödinger equations. Our system under consideration is a generalization of the equations which follow from the analysis of optical fibres. Reductions of some special equations are given.     

Eigenvalue Bounds for Schrödinger Operators with a Homogeneous Magnetic Field

We prove Lieb-Thirring inequalities for Schrödinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.     

Measure Zero Spectrum of a Class of Schrödinger Operators

We study the measure of the spectrum of a class of one-dimensional discrete Schrödinger operators H _v, with potential v() generated by any primitive substitutions. It is well known that the spectrum of H _v, is singular continuous.⁽¹⁾ We will give a more exact result that the spectrum of H _v, is a set of Lebesgue measure zero, by removing two hypotheses (the semi-primitive of a certain induced substitution and the existence of square word) from a theorem due to Bovier and Ghez.⁽²⁾     

Asymptotic Number of Scattering Resonances for Generic Schrödinger Operators

Let ?Δ + V be the Schrödinger operator acting on ${L^2(\mathbb{R}^d,\mathbb{C})}$ with ${d\geq 3}$ odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. Let n _V (r) denote the number of resonances of ?Δ + V with modulus ≤  r. We show that if the potential V is generic in a sense of pluripotential theory then $$n_V(r)=c_d a^dr^d+ O(r^{d-{3\over 16}+\epsilon}) \quad \mbox{as } r \to \infty$$ for any ε > 0, where c _d is a dimensional constant.     

Symmetries of Separating Nonlinear Schrödinger Equations

Abstract

We review here the main properties of symmetries of separating hierarchies of nonlinear Schrödinger equations and discuss the obstruction to symmetry liftings from (n)-particles to a higher number. We argue that for particles with internal degrees of freedom, new multiparticle effects must appear at each particle-number level.     

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

Schr?dinger Dispersive Estimates for a Scaling-Critical Class of Potentials

We prove a dispersive estimate for the evolution of Schr?dinger operators H = ??? + V(x) in ${{\mathbb R}^3}$ . The potential should belong to the closure of ${C^c_b(\mathbb{R}^3)}$ with respect to the global Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere within the interval [0, ??). The proof is an application of a new version of Wiener??s L ¹-inversion theorem.     

Bounds on the number of eigenvalues of the Schrödinger operator

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:

1. A connection between the moments of order (n ? 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN _n, inn space dimensions;
2. optimal bounds on the total number of bound states below a given energy in one dimension;
3. alower bound onN ₂;
4. a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC ₀₄,C ₃;
5. solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC _0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.

     

Bounds on the density of states of random Schrödinger operators

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=?Δ + V acting onL ²(? ^d ) orl ²(? ^d ). In both cases the random potential is $$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$ in which the \(\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }\) areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the \(\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }\) are cells of unit dimensions centered ony∈? ^d . In the finite-difference case Λ(y) denotes the sitey∈? ^d itself. Under the assumptionf ∈ L ₀ ^1+? (?) it is proven that in the finitedifference casep ∈ L ^∞(?), and that in thed= 1 continuum casep ∈ L _loc ^∞ (?).     

New Spherically Symmetric Solutions of Nonlinear Schrödinger Equations

Abstract

New soliton-like spherically symmetric solutions for nonlinear generalizations of the Schrödiner equation are constructed. A new nonlinear projective invariant Schrödiner equation is suggested and formulae of multiplication of its solutions are found.     

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed.