Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

Superpropagators of non-localizable fields

Arguments are given for a special choice of the superpropagator corresponding to Wightman-two-point-functions with an increase in momentum space as , >0.     

Layering transition in SOS model with external magnetic field

For the SOS model defined by the Hamiltonian , where _x , _x ,{1,2,...},h>0,x ^d ,d2 it is shown that in the low-temperature region an infinite sequence of first-order phase transitions takes place whenh»0 and the temperature is fixed.     

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field and static magnetic field for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as , where , , and . Here m^* and e denote the effective mass and charge of electron, respectively. _g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear v₀ vs E₀ characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

Level-rank duality of WZW models in conformal field theory

We consider the decomposition of the conformal blocks under the conformal embeddings. The case (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of -modules induce a pairing between the spaces of conformal blocks of and Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.Dedicated to Professor Masahisa Adachi on his 60th birthday     

Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I

We consider the Zakharov equation in space dimension two

Periodic nonlinear Schrödinger equation and invariant measures

In this paper we continue some investigations on the periodic NLSEiu _u +iu _xx +u|u| ^p-2 (p6) started in [LRS]. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measrue (after suitableL ²-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.     

Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. Part II

We consider the Zakharov equation in space dimension two

Symmetry breaking in Landau gauge a comment to a paper by T. Kennedy and C. King

For the non-compact abelian lattice Higgs model in Landau gauge Kennedy and King (Princeton preprint, 1985) showed that the two point function does not decay in the Higgs phase. We generalize their methods to show that for the same range of parameters there are states parametrized by an angle [0, 2) such that and 0$$ " align="middle" border="0"> .     

On the Modified Korteweg–De Vries Equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation , with initial data . We assume that the coefficient is real, bounded and slowly varying function, such that , where . We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space . In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates and for all , where . We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.     

Uncertainty Relation on Generalized Skew Information with aMonotone Pair

In this paper, we first define a generalized (f,g)-skew information \(\left |I_{ \rho }^{(f, g)}\right |(A)\) and two related quantity \(\left |J_{ \rho }^{(f, g)}\right |(A)\) and \(\left |U_{ \rho }^{(f, g)}\right |(A)\) for any non-Hermitian Hilbert-Schmidt operator A and a density operator ρ on a Hilbert space H and discuss some properties of them. And then, we obtain the following uncertainty relation in terms of \(\left |U_{ \rho }^{(f, g)}\right |(A)\):

$$\begin{array}{@{}rcl@{}} \left|U_{ \rho}^{(f, g)}\right|(A)\left|U_{ \rho}^{(f, g)}\right|(B)\geq \beta_{(f, g)}\left|Tr\left( f(\rho)g(\rho)[A, B]^{0}\right)\right|^{2}, \end{array} $$

which is a generalization of a known uncertainty relation in Ko and Yoo (J. Math. Anal. Appl. 383, 208–214, 11).

     

Tunneling Across an Inhomogeneous Delta-Barrier

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy (where 0$$ " align="middle" border="0"> and 0$$ " align="middle" border="0"> are two constants). In particular, the perpendicular incidence of an electron with a given value of the wave vector is considered. The electron is forward-scattered into the region behind the barrier (region 2: 0$$ " align="middle" border="0"> ), i. e. the wave function is composed of plane waves with all wave vectors such that and \left. 0 \right)} $$ " align="middle" border="0"> ) (where ). Therefore, if 0$$ " align="middle" border="0"> , the wave function of the electron is represented as , where . An approximate formula is derived for the amplitude . The authors pay a special attention to the flow density and calculate this function in two cases: 1. for the plane and 2. for high values of is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.     

On the Davey–Stewartson and Ishimori Systems

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations i u_t + u = N(v), (t, x, y) R³, u(0, x, y) = u₀(x, y), (x, y) R² (A), where the Laplacian = ² _x + ² _y, the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁(v) which is cubic with respect to the vector v=(u,u_x,u_y,\overline{u},\overline{u}_{x},\overline{u}_{y}) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) =(v, ^{– 1} _x K_x(v)) + (v, ^{– 1} _y K_y(v)), where (, ) denotes the inner product, and the vectors K_x (C⁴(C⁶; C))⁶ and K_y (C⁴(C⁶; C))⁶ are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components K⁽²⁾ _x = K⁽⁴⁾ _x 0, K⁽³⁾ _y = K⁽⁶⁾ _y 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with , and all the rest components of the vectors K_x and K_y are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when , and . Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N(eⁱ v) = eⁱ N(v) for all R.     

Unitary dressing transformations and exponential decay below threshold for quantum spin systems. Parts III and IV

This is the continuation of a series of articles concerning a class of quantum spin systems with Hamiltonian operators of the form

Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)$$ and that $$\mathop {\lim \sup }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.     

Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation

Yukawa coupling of quantum fields in two dimensions. I

A renormalization procedure is proposed. It gives rigorous mathematical meaning to the infinite cancellations in this model. A space cutoff is introduced in the interaction termV and soV has the form , but there are no momentum cutoffs inV. There is an infinite constant and an infinite boson mass renormalization in this model. The main result is that the renormalized Hamiltonian is rigorously defined as a bilinear form in the Fock Hilbert space.This work was supported in part by the National Science Foundation, GP-6165.     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

Integrals of Vertex Operators and uantum Shuffles

Given a braided vector space , we show that iterated integrals of operator-valued functions satisfying a certain exchange relation give rise to representations of the quantum shuffle algebra built on . Using the quantum shuffle construction of the 'upper triangular part' of a quantum shuffle, this provides a simple proof of the result of Bouwknegt, MacCarthy and Pilch saying that integrals of vertex operators acting on certain Fock modules give rise to representations of .     

A Ruelle operator for a real Julia set

LetR be an expanding rational function with a real bounded Julia set, and let be a Ruelle operator acting in a space of functions analytic in a neighbourhood of the Julia set. We obtain explicit expressions for the resolvent function and, in particular, for the Fredholm determinantD()=det(I-L). It gives us an equation for calculating the escape rate. We relate our results to orthogonal polynomials with respect to the balanced measure ofR. Two examples are considered.The first named author was sponsored in part by the Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany)