The construction of the eigenstates for nonlinear Schrödinger model with supermatrices and attractive coupling

A quantum nonlinear Schrödinger model with supermatrices and attractive coupling is studied by using the quantum inverse scattering method. The eigenstates of the Hamiltonian and the infinite number of the conserved quantities of the system are constructed. In particular, theN-particle bound states with the mixture of bosons and fermions are found. The energy of theN-particle eigenstate are Σ _i=1 ^N andNp ² ?N(N ²?1)c ²/12 for the scattering state and the bound state respectively.     

Quantum hypervirial theorems

The quantum mechanical hypervirial theorems (HVT) are given by the expectation values of a commutator between the virial operator W and the system Hamiltonian H. We propose application of the HVT to testing and improving approximate solutions of the Schroedinger equations. This is especially relevant for scattering states, where simple testing criteria are not readily available. The HVT, with judicious choices for W, can provide a criterion to test the accuracy of approximate solutions, both for the bound excited states and scattering states, and also ways to determine an optimal set of parameter values as the wave function improves.     

Yangian代数在混合介子态的纠缠和衰变中的应用

Yangian代数是超出李代数更大的无穷维代数,是研究非线性量子完全可积系统的新对称特性的有力数学工具.基于介子态中夸克-味su(3)对称性和Yangian代数生成元的跃迁特性,本文研究了Yangian代数Y(su(3))生成元在三种正反介子态(π~±,K~±,K~0和K~0)各自组成的三种混合介子态(π,K和K_i~0)衰变中的作用.将Y(su(3))代数的八个生成元(I~±,U~±,V~±,I~3和I~8)作为跃迁算子,作用在混合介子态上,研究其可能的衰变道,以及衰变前后纠缠度的变化.结果表明:1)在李代数范围内的生成元I~3和I~8作用下,三种混合介子态衰变后组成成分没有发生变化,其中混合介子态π在I~8作用下衰变前后纠缠无变化,其他衰变纠缠度发生了变化;2)在其他的六个(I~±,U~±和V~±)超出李代数的生成元的作用下,三种混合介子态衰变前后组成成分发生了变化,其中两个衰变后变成单态,纠缠度为零;两个衰变不存在;剩余两个衰变后纠缠度发生了变化,此外在带电(K)和中性(K_I~0)两类K型混合介子态的六种可能的衰变中,两种类型的末态的纠缠度两两相同;3)三种混合介子态之间可以通过I~±,U~±和V~±算子循环转化,具有明显的对称性.本文从具有的对称性上提供了一种探索混合介子态可能衰变的方法,并且可以用此方法去预测可能的未知衰变粒子和解释己测得的衰变问题.     

Spatially inhomogeneous states of charge carriers in graphene

The interaction of two-dimensional quasiparticles characterized by a linear dispersion E = ±u|p| (graphene) with impurity potentials is studied. It is shown that discrete levels corresponding to localized states are present in a one-dimensional potential well (quantum wire), whereas such states are absent in a two-dimensional well (quantum dot). The cross section for the scattering of electrons (holes) of graphene by an axially symmetric potential well is determined. It is shown that the cross section tends to a constant value in the limit of infinite particle energy. The effective Hamiltonian is derived for a curved quantum wire of graphene.     

Testing QCD trough inclusiveJ/ψ-production ine +e− annihilations

We study the cubic Schrödinger model for attractive coupling. Using the quantized version of the Zakharov-Shabat eigenvalue problem we define operators, which create the eigenstates of the corresponding quantum field theoretical Hamiltonian. In particular, we construct the well-knownn-particle bound states of this model.     

最一般的超矩阵量子非线性Schr?dinger模型的本征态研究

本文用量子反散射方法讨论了最一般的超矩阵量子非线性Schr?dinger模型的本征态。特别是构造了玻色和费密粒子混合的N粒子束缚态。N粒子的散射态和束缚态的能量分别为∑_j=1^Nλ_j²和Np²-(c²/12)N(N²-1)。 关键词：     

General solution of relativistic one-channel inverse scattering problem

We find explicitly in the p-representation the kernels of the logarithms of unitary operators transforming the free Hamiltonian into the general solution of the one-channel inverse scattering problem, when bound states are absent. Then we construct the transformation of the Hamiltonian adding to its spectrum the given set of bound states without changing the scattering operator. Other generators of the Poincaré group are constructed in a similar way. The proof of some relevant limits is given.     

Application of Y（sl（2）） Algebra for Entanglement of Two-Qubit System

We construct the transition operators in terms of the generators of the general Yangian and the reduced Yangian. By acting these operators on a two-qubit pure state, we find that the entanglement degrees of the states are all decreased from the certain values to zero for the reduced Yangian algebra, which makes the state disentangled. This result sheds new light on the physical meaning of Y （sl（2） ） in quantum information.     

Metastable States in Parametrically Excited Multimode Hamiltonian Systems

 Consider a linear autonomous Hamiltonian system with m time periodic bound state solutions. In this paper we study their dynamics under time almost periodic perturbations which are small, localized and Hamiltonian. The analysis proceeds through a reduction of the original infinite dimensional dynamical system to the dynamics of two coupled subsystems: a dominant m-dimensional system of ordinary differential equations (normal form), governing the projections onto the bound states and an infinite dimensional dispersive wave equation. The present work generalizes previous work of the authors, where the case of a single bound state is considered. Here, the interaction picture is considerably more complicated and requires deeper analysis, due to a multiplicity of bound states and the very general nature of the perturbation's time dependence. Parametric forcing induces coupling of bound states to continuum radiation modes, of bound states directly to bound states, as well as coupling among bound states, which is mediated by continuum modes. Our analysis elucidates these interactions and we prove the metastability (long life time) and eventual decay of bound states for a large class of systems. The key hypotheses for the analysis are: appropriate local energy decay estimates for the unperturbed evolution operator, restricted to the continuous spectral part of the Hamiltonian, and a matrix Fermi Golden rule condition, which ensures coupling of bound states to continuum modes. Problems of the type considered arise in many areas of application including ionization physics, quantum molecular theory and the propagation of light in optical fibers in the presence of defects. Received: 13 March 2002 / Accepted: 2 January 2003 Published online: 14 April 2003 Communicated by J.L. Lebowitz     

On-shell faddeev equations: A democritean approach to the three-body problem

Coupling the mass-energy relationδE≧mc² to the uncertainty relationδE δt ≧ ? produces fluctuations in the number of particles at short distances and scatterings of particle pairs independent of any specific “interaction” mechanism. This observation allows the construction of a scattering theory in which there are only particles and the void, but particle number can change. We consider a system of three massive particles (hadrons) in the energy region below the first production threshold for a fourth hadron and above the first anomalous threshold for the presence of a fourth “virtual” hadron. The on-shell Faddeev equations, containing only two-particle scattering phases for positive two particle energies, provide a convergent, unitary, and readily soluble dynamics for this system. If any of the pairs can coalesce into a different particle with a rest energy less than the sum of the rest energies of the pair, the equations can be readily extended to describe 3-2 and 2–3 transitions involving this particle (coalescence, breakup) elastic scattering from it, and if there is more than one such particle 2-2 rearrangements. The three-body “bound state” requires a well defined analytic continuation. Features of more conventional calculations of three-nucleon problems which provide examples of this structure are discussed. Since only free particles occur in the theory, and the only failure of energy conservation is that required by the uncertainty principle for (free-particle) intermediate states, these one-variable equations might be extended to particles with the relativistic connection between mass, energy and momentum, and transitions in which the full rest energy of the particle which appears or disappears must be provided. The non-linear “crossed” theory for such particles has not been written down, but if the relativistic boundary condition model of Brayshaw is viewed as representing these crossed processes by a phenomenological core, then a crossed theory requiring the π to be a bound state of three π's might predict the π-π S-wave scattering length in theI=0 state in terms of the pion Compton wavelength (and hence the position and the width of the?) and will then show that the? in turn generates asingle ω resonance at about the right place. Implications are discussed.     

Classification of reflection matrices related to (super-)Yangians and application to open spin chain models

We present a classification of diagonal, antidiagonal and mixed reflection matrices related to Yangian and super-Yangian R matrices associated to the infinite series so(m), sp(n) and osp(m|n). We formulate the analytical Bethe ansatz resolution for the so(m) and sp(n) open spin chains with boundary conditions described by the diagonal solutions.     

The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμ_λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm-Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ_m,n and the energies E_m,n of the bound states are exactly obtained in both the sphere S² and the hyperbolic plane H².     

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U _q (sl(n)), Y(sl(n)) (the Yangian) and U _q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo     

Nonlinear Raman scattering of photons by a channeled particle

Channeled particles are characterized by the discrete spectrum of bound transverse motion. The interaction of photons with channeled particles in single crystals can be accompanied by energy transitions between the levels of transverse motion of the channeled particle. The Raman scattering of photons at a quasibound channeled particle leads to the appearance of a combination of frequencies: the incident radiation frequency ω₀ and the frequency Δω_{m, n}, i.e., ω = ω₀ ± Δω_m,n where Δω_m,n = 2Δε_m,nγ²; Δε_{m, n} is the energy of the transition between quantum states (m and n) of the transverse motion of the channeled particle; and γ = E/mc² is the Lorentz factor of the channeled particle. The appearance of a violet satellite (the anti-Stokes component) in the Raman scattering spectrum is analyzed. The three-photon Raman-type transition, which is the process of the simultaneous absorption of two photons with the frequency ω₀ with the emission of a photon with the frequency ωs = 2ω₀ ± 2Δε_m,nγ², is considered. The conditions for resonance observation during the formation of the second harmonic (ω = 2ω₀) are discussed.     

A novel strong coupling expansion of the QCD Hamiltonian

Introducing an infinite spatial lattice with box length a, a systematic expansion of the physical QCD Hamiltonian in λ = g ^?2/3 can be obtained, with the free part being the sum of the Hamiltonians of the quantum mechanics of spatially constant fields for each box, and interaction terms proportional to λ ⁿ with n spatial derivatives connecting different boxes. As an example, the energy of the vacuum and the lowest scalar glueball is calculated up to order λ ² for the case of SU(2) Yang-Mills theory.     

Transfer of a weakly bound electron in collisions of Rydberg atoms with neutral particles. II. Ion-pair formation and resonant quenching of the Rb(nl) and Ne(nl) States by Ca, Sr, and Ba atoms

Electron-transfer processes are studied in thermal collisions of Rydberg atoms with alkaline-earth Ca(4s ²), Sr(5s ²), and Ba(6s ²) atoms capable of forming negative ions with a weakly bound outermost p-electron. We consider the ion-pair formation and resonant quenching of highly excited atomic states caused by transitions between Rydberg covalent and ionic terms of a quasi-molecule produced in collisions of particles. The contributions of these reaction channels to the total depopulation cross section of Rydberg states of Rb(nl) and Ne(nl) atoms as functions of the principal quantum number n are compared for selectively excited nl-levels with l ? n and for states with large orbital quantum numbers l = n ? 1, n ? 2. It is shown that the contribution from resonant quenching dominates at small values of n, and the ion-pair formation process begins to dominate with increasing n. The values and positions of the maxima of cross sections for both processes strongly depend on the electron affinity of an alkaline-earth atom and on the orbital angular momentum l of a highly excited atom. It is shown that in the case of Rydberg atoms in states with large l ～ n ? 1, the rate constants of ion-pair formation and collisional quenching are considerably lower than those for nl-levels with l ? n.     

Scattering from correlated systems

A difficulty usually encountered in formulating the problem of scattering of identical particles from correlated systems is that the customary choice of an unperturbed Hamiltonian as the target Hamiltonian plus the kinetic energy of the projectile is not symmetric under particle exchange. This choice of unperturbed Hamiltonian leads to wavefunctions which, if they are antisymmetrized, are not orthonormal. In this paper an orthonormal, antisymmetrized set of basis states is constructed. These states are then used to construct a symmetric unperturbed Hamiltonian, so that a formal scattering equation with appropriate boundary conditions can be written. An expression for a T matrix describing nucleon-nucleus scattering can then be obtained. The formalism leads to a two-potential form for the T matrix, the first term of which describes the effect of the orthogonality of the scattering state and the negative energy states.     

On the integrals of motion for an exactly solvable model of interacting fermions

Based on the Lax formalism, integrals of motion are constructed for the Sutherland hyperbolic systems of particles with internal degrees of freedom (su(n) spins) situated in an external field with the Morse potential characterized by the parameter τ². It is shown that the corresponding infinite-dimensional algebra determining the hidden symmetry of the systems is not of the Yangian type.     

Bound state approach for strange dibaryons in the skyrme model

The bound state approach to strange dibaryons in the Skyrme model is extended to baryon number n > 1. Kaon bound states are obtained in a (variational) axially symmetric SU (2) skyrmion background field. Collective quantization of isospin and spatial zero modes leads to dibaryon quantum states. These are classified in flavor multiplets.