Action–angle variables,ladder operators and coherent states

This Letter is devoted to the building of coherent states from arguments based on classical action–angle variables. First, we show how these classical variables are associated to an algebraic structure in terms of Poisson brackets. In the quantum context these considerations are implemented by ladder type operators and a structure known as spectrum generating algebra. All this allows to generate coherent states and thereby the correspondence of classical–quantum properties by means of the aforementioned underlying structure. This approach is illustrated with the example of the one-dimensional Pöschl–Teller potential system.     

Squeezed “atomic” states,pseudo-Hermitian operators and Wigner D-matrices

The states of N two-level atoms can be mapped onto the eigenvectors of angular momentum (with j=N/2) and this system in interaction with a radiation field constitutes a fundamental model in Quantum Optics. There from one may construct atomic coherent states and minimum uncertainty packets. The squeezing of such states is of considerable contemporary interest. We show that the properties of squeezed atomic states are most elegantly and economically expressed in terms of pseudo-Hermitian operators and through Wigner D-matrices and their analytical continuation.     

Feynman path integrals and the trace formula for the Schrödinger operators

We study Schrödinger operators of the form on ^d , whereA ² is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If _n are the eigenvalues ofH we show that the theta function is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of (t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt Unendlich dimensionale Analysis)     

Expanding and calculation of high-order uncommutation operators

A general method has been suggested for expanding the high-order uncommutation operators and developing several formulations of their series expression. Then the expanding and calculation of Poisson bracket which consist of high-order operations should be more convenient and direct.     

Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and of non-linear Schrödinger equations

We consider, in a 1+3 space time, arbitrary (finite) systems of nonlinear Klein-Gordon equations (respectively Schrödinger equations) with an arbitrary local and analytic non-linearity in the unknown and its first and second order space-time (respectively first order space) derivatives, having no constant or linear terms. No restriction is given on the frequency sign of the initial data. In the case of non-linear Klein-Gordon equations all masses are supposed to be different from zero.We prove, for such systems, that the wave operator (fromt= tot=0) exists on a domain of small entire test functions of exponential type and that the analytic Cauchy problem, in ⁺׳, has a unique solution for each initial condition (att=0) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.Partially supported by the Swiss National Science FoundationOn leave from Institut de Physique Théorique, 32 Bd d'Ivoy, CH-1211 Geneve 4 Switzerland.     

A semi-classical trace formula for Schrödinger operators

LetS _?=??Δ+V, withV smooth. If 0<E ²V(x), the spectrum ofS _? nearE ² consists (for ? small) of finitely-many eigenvalues,λ _j (?). We study the asymptotic distribution of these eigenvalues aboutE ² as ?→0; we obtain semi-classical asymptotics for $$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$ with \(\hat f \in C_0^\infty \) , in terms of the periodic classical trajectories on the energy surface \(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\) . This in turn gives Weyl-type estimates for the counting function \(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\) . We make a detailed analysis of the case when the flow onB _E is periodic.     

Singular continuous spectrum for palindromic Schrödinger operators

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x _n =1 _J (₀+n), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational and every half-open intervalJ.Work partially supported by NSERC.This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material.     

On eigenvalue estimates for the Dirac–Witten-type operators

We give optimal lower bounds for the eigenvalues of the Dirac–Witten-type operators associated with the _e0$e_0$-Killing connection and imaginary Killing connection, in terms of the mean curvature and the scalar curvature. The limiting cases are then studied and lead to interesting geometric situations.     

Localization for some continuous random Schrödinger operators

We study the spectrum of random Schrödinger operators acting onL ²(R ^d ) of the following type . The are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.

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Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL ²(R ^d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.

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U.R.A. 760 C.N.R.S.     

Photoionisation of atoms and Möller operators

The motion of an hydrogenoïd atom in a laser field is usually given by the time-dependent hamiltonian H(t)=[p?A(t)]²/2+V(r) where V(r) is the atomic potential whileA(t) is to be connected with the laser field. The existence and unicity for the Cauchy problem of the solutions of the corresponding Schrödinger equation are established under mild conditions onA(t) and V(r). The existence of Möller operators is investigated in two cases, namely, when the laser field is a function of time only and when it vanishes asymptotically in time. Special attention is paid for the Coulomb case for which a “distorted” Möller operator is derived. Finally, when the laser field vanishes ast→∞, the photoionisation probability is properly defined by means of the Möller operator $$\Omega (H_{At} ,H) = s - \mathop {\lim }\limits_{t \to \infty } U_{At} (t)^{ - 1} U(t)$$ , whereU(t) is the evolution operator for the system whileU _Att (t) is the evolution operator for the atom.     

Born series for (2 cluster) → (2 cluster) scattering of two,three, and four particle Schrödinger operators

We investigate elastic and inelastic (2 cluster) (2 cluster) scattering for classes of two, three, and four body Schrödinger operators Formulas are derived for those generalized eigenfunctions ofH which correspond asymptotically in the past to two freely moving clusters. With these eigenfunctions, we establish a formula for the (2 cluster) (2 cluster)T-matrix and prove the convergence of a Born series for theT-matrix at high energy.Supported in part by the National Science Foundation under Grant PHY 78-08066     

Localization for random Schrödinger operators with correlated potentials

We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|^–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627     

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator ;f0,q _i ()0, has Lifschitz singularities at the edges of the gaps inSp(H ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.     

Magnon–magnon interactions in O(3) ferromagnets and equations of motion for spin operators

The method of equations of motion for spin operators in the case of O(3) Heisenberg ferromagnet is systematically analyzed starting from the effective Lagrangian. It is shown that the random phase approximation and the Callen approximation can be understood in terms of perturbation theory for type B magnons. Also, the second order approximation of Kondo and Yamaji for one dimensional ferromagnet is reduced to the perturbation theory for type A magnons. An emphasis is put on the physical picture, i.e. on magnon–magnon interactions and symmetries of the Heisenberg model. Calculations demonstrate that all three approximations differ in manner in which the magnon–magnon interactions arising from the Wess–Zumino term are treated, from where specific features and limitations of each of them can be deduced.     

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ ₁,…,δ _m ) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.     

Conservation laws,modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrödinger equation, which describes the pulses propagating along an optical fiber. Based on the symbolic computation, Lax pair and infinitely-many conservation laws are derived. Via the modiied Hirota method, bilinear forms and multi-soliton solutions are obtained. Propagation and interactions of the solitons are illustrated graphically: Initial position and velocity of the soliton are related to the coefficient of the sixth-order dispersion, while the amplitude of the soliton is not affected by it. Head-on, overtaking and oscillating interactions between the two solitons are displayed. Through the asymptotic analysis, interaction between the two solitons is proved to be elastic. Based on the linear stability analysis, the modulation instability condition for the soliton solutions is obtained.