No abstract

No abstract

No abstract

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit.As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k.For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k D 0, which yields an asymptotic formula for the reflection coefficient at k D 0 and suggests an annular structure for the solution that may be exploited when k ¤ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as # 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.

No abstract

It is shown that if Φ is a finite range interaction of a quantum spin system, τf the associated group of time translations, τ x the group of space translations, and A, B local observables, then lim ||[τ?τ 3e (A),B]||e" (l ' )t =0 \*\>v\t\ whenever v is sufficiently large (v > V φ ) where μ(υ) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514

This paper has three aims: (i) To discuss some of the mathematical connections between N‐particle wave functions ψ and their single‐particle densities ρ (x). (ii) To establish some of the mathematical underpinnings of “universal density functional” theory for the ground state energy as begun by Hohenberg and Kohn. We show that the HK functional is not defined for all ρ and we present several ways around this difficulty. Several less obvious problems remain, however. (iii) Since the functional mentioned above is not computable, we review examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy.