Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation

Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation are compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral stability. Small eigenvalues bifurcating from the zero eigenvalue near the anti-continuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out the existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) near the anti-continuum limit.     

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.     

Quantum mechanics in discrete space and angular momentum

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

In this paper we describe invariant geometrical structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable. Received: 30 January 1997 / Accepted: 6 April 1997     

Remark on the (non)convergence of ensemble densities in dynamical systems

We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.     

Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant

We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the lambda --> infinity limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant.     

Resonance conditions for the spectral function and single-hole states in finite nuclei

An equation for the spectroscopic amplitudes in finite nuclei is determined both in the discrete and in the continuous spectrum of the residual nucleus hamiltonian. Its structure, involving the hole mass operator, is investigated. In the continuous spectrum, the spectroscopic amplitude equation leads to a precise definition of the hole decay width (related to the average value of the antihermitian part of the hole mass operator). Besides, one deduces a formula which describes the resonant behaviour of the spectral function in terms of the hole decay widths. Finally, the resonance conditions are investigated. Close to sharp resonances (included the ones of the discrete spectrum), one interpretes the spectroscopic amplitudes as single-hole wave functions which satisfy a Schrödinger-like equation where the hermitian part of the hole mass operator plays the role of an effective potential.     

Study of the process e+e-→π+π-π0π0 at energies √s< 1 GeV

In an experiment with the Spherical Neutral Detector at VEPP-2M collider the cross section of the process e+e-→π+π-π0π0 was measured. At energies √s < 920 MeV this cross section was measured for the first time. The energy dependence of the cross section is well discribed by the vector dominance model with contributions from ρ, ρ' ρ", mesons. The decay probability ρ→π+π-π0π0was found to be Bρ = (1.60±0.74±0.18)×10-5. The upper limit for the decay ω→π+π-π0π0 was improved by two orders of magnitude compared to the previous measurements and is Bω < 2 × 10-4 at 90% confidence level.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Periodic Waves of a Discrete Higher Order Nonlinear SchrSdinger Equation

The Hirota equation is a higher order extension of the nonlinear Schr6dinger equation by incorporating third order dispersion and one form of self steepening effect, New periodic waves for the discrete Hirota equation are given in terms of elliptic functions. The continuum limit converges to the analogous result for the continuous Hirota equation, while the long wave limit of these discrete periodic patterns reproduces the known resulr of the integrable Ablowitz-Ladik system.     

Equation of motion for a string operator in two-dimensional massless lattice QCD

In two-dimensional massless lattice QCD we construct the equation of motion for a gauge invariant meson operator with a quark-antiquark pair connected by the path-ordered products of link variables. Using the large-N factorization property the meson wave equation is derived ain a nearly identical form to the 't Hooft equation.     

Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ ₀ liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity.     

Higher-Dimensional Lie Algebra and New Integrable Coupling of Discrete KdV Equation

Based on semi-direct sums of Lie subalgebra \tilde{G}, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derivedfrom a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity --- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we provethat the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.     

Radiative decays of ρ and ω mesons

The results of the measurements of radiative decays of ρ and ω mesons with the Neutral Detector at thee ⁺ e ^? collider VEPP-2M are presented. The branching ratio of the decay ω→π ⁰γ was measured with higher than in previous experiments accuracy: $${\rm B}(\omega \to \pi ^0 \gamma ) = 0.0888 \pm 0.0062$$ . The ρ⁰→π ⁰ γ branching ratio was measured for the first time: $$B(\rho ^0 \to \pi ^0 \gamma ) = (7.9 \pm 2.0) \cdot 10^{ - 4} $$ . The decays ρ, ω→ηγ were studied. Their branching ratios with the assumption of constructive ρ?ω interference are: $$\begin{gathered} B(\omega \to \eta \gamma ) = (7.3 \pm 2.9) \cdot 10^{ - 4} , \hfill \\ B(\rho \to \eta \gamma ) = (4.0 \pm 1.1) \cdot 10^{ - 4} \hfill \\ \end{gathered} $$ . The branching ratios of ρ, ω→ηγ and ω→e ⁺ e ^? decays were also measured: $$\begin{gathered} B(\omega \to \pi ^ + \pi ^ - \pi ^0 ) = 0.8942 \pm 0.0062, \hfill \\ B(\omega \to e^ + e^ - ) = (7.14 \pm 0.36) \cdot 10^{ - 5} \hfill \\ \end{gathered} $$ . The upper limit for the ω→π ⁰ π ⁰ γ branching ratio was placed: B(ω→π ⁰ π ⁰ γ)<4·10^?4 at 90% confidence level.     

Perturbation approach to plasma oscillations

It is shown that the complete spectrum, continuous as well as discrete, and corresponding eigenfunctions of the Vlasov operator can be obtained by a single perturbation procedure from the purely continuous spectrum and corresponding eigenfunctions of the free-streaming operator. In addition we present an alternative definition of the eigenfunctions and show that the problem of normalizing the continuous eigenmodes is thus solved automatically.     

Aspects of Solutions of Massive Spin-3/2 Equation in Schwarzschild Space-Time

The general massive spin-(3/2) (Rarita–Schwinger) field equation in Schwarzschild geometry, previously separated by variable separation, is further studied. The orthogonality of the solutions of the angular equations is exploited. The study of the radial equations, that are proposed in the most detailed form, is reduced to the study of four coupled differential equations. The equations are discussed and integrated near the Schwarzschild radius and for zero and large values of the radial coordinate. A covariant product of states is considered that is induced by a conserved current. It is shown the existence of states that are bound in the scalar product without implying the existence of a discrete energy spectrum.     

Coherent states of the inverted Caldirola-Kanai oscillator with time-dependent singularities

The coherent states for a system of time-dependent singular potentials coupled to inverted CK (Caldirola-Kanai) oscillator are investigated by employing invariant operator method and Lie algebraic approach. We considered Coulomb potential and inverse quadratic potential as singularities of the system. The spectrum of quantum states is discrete for λ < 0 while continuous for λ ? 0. The probability densities for both Fock state and coherent state are converged to the center as time goes by according to the dissipation of energy. We confirmed that the probability density in the coherent state oscillates back and forth like a classical wave packet.