Invariant Bidifferential Operators on Tensor Densities over a Contact Manifold

The spaces of tensor densities over a manifold M are modules over the Lie algebra Vect(M) of vector fields over the manifold. When M is a contact manifold, one can consider the algebra C(M) of vector fields which preserves the contact structure. If the manifold is endowed with a contact projective structure, there is an embedding of the linear symplectic algebra sp(2n+2, ) in C(M). In this Letter, we determine the C(M)- and the sp(2n+2, )-invariant bidifferential operators on tensor densities.     

Odd Invariant Semidensity and Divergence-like Operators on an Odd Symplectic Superspace

The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hypersurfaces in Euclidean space. Received: 19 August 1997 / Accepted: 27 March 1998     

Projectively Equivariant Quantization for Differential Operators Acting on Forms

In this Letter, we show the existence of a natural and projectively equivariant quantization map depending on a linear torsion-free connection for the spaces of differential operators mapping p-forms into functions on an arbitrary smooth manifold M. We show how this result implies the existence over of an sl _m+1-equivariant quantization for the spaces .This revised version was published online in March 2005 with corrections to the cover date.     

Equivariant Symbol Calculus for Differential Operators Acting on Forms

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces of differential operators transforming p-forms into functions, over . As an application, we classify the Vect(M)-equivariant maps from to over a smooth manifold M, recovering and improving earlier results of N.Poncin. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

Variational Operators,Symplectic Operators,and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation u_t = K(t, x, u, u_x, . . . , u_2m+1) with m ≥ 1, the cohomology space H^1,2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u_t = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H^1,2() and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

Transformations on Density Operators That Leave the Holevo Bound Invariant

For a given probability distribution λ ₁,…,λ _m we determine the structure of all such maps defined on a dense subset of density operators which leave the Holevo bound invariant i.e. which satisfy $$S\Biggl(\sum_{k=1}^m \lambda_k \phi(\rho_k)\Biggr)- \sum_{k=1}^m \lambda_k S\bigl(\phi (\rho_k)\bigr)= S\Biggl(\sum _{k=1}^m \lambda_k \rho_k\Biggr)- \sum_{k=1}^m \lambda_k S(\rho_k) $$ for all possible collections ρ ₁,…,ρ _m of density operators.     

Tensor Fields on Self-Dual Warped AdS_3 Background

Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3-dimensional self-dual space-like warped AdS_3 black hole background. We obtain a Klein–Gordon-like equation for tensor fields. By using gauge constraint and traceless condition, we will find the exact solutions of the equations of motion. Then, we will compute the quasi-normal modes by imposing appropriate boundary conditions at horizon and infinity.     

Note on Energy-Momentum Tensor for General Mixed Tensor-Spinor Fields

This note provides an explicit proof of the equivalence of Belinfante＇s energy-momentum tensor and metric energy-momentum tensor for general mixed tensor-spinor fields.     

Note on Energy-Momentum Tensor for General Mixed Tensor-Spinor Fields

This note provides an explicit proof of the equivalence of Belinfante‘s energy-momentum tensor and metric energy-momentum tensor for general mixed tensor-spinor fields.     

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence, and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.     

强激光场中CO分子经典轨迹的辛算法

采用辛算法计算CO异核双原子分子系统在强激光场作用下的经典轨迹,与Runge-Kutta方法进行比较,分析了CO双原子分子在激光场作用下的振动轨迹、相平面轨道与总能量随时间的变化及CO分子的解离.     

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

On the Spectrum of a Class of Second Order Periodic Elliptic Differential Operators

Under an additional symmetry condition, we prove that the spectrum of a second order self-adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous. Received: 19 December 2001 / Accepted: 10 January 2002?Published online: 24 July 2002     

Noncommutative Differential Forms and Quantization of the Odd Symplectic Category

There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relations) in terms of ₂-graded chain complexes. It is a straightforward consequence of the theory of BV operator acting on semidensities, due to H. Khudaverdian.     

Connectedness of the Isospectral Manifold for One-Dimensional Half-Line Schrödinger Operators

Journal of Statistical Physics - Let V 0be a real-valued function on [0,∞) and V∈L 1([0,R]) for all R&;gt;0 so that H(V 0)=? $\frac{{d^2 }}{{dx^2 }}$ +V 0in L 2([0,∞))...     

Existence and Completeness of the Wave Operators for Higher Order Differential Operators

The aim of the present paper is to study the existence and the completeness of the wave operators for elliptic operators of higher order (Schr?dinger operator) with short-range potential of the form: and other related results by using the trace class method. MSC: 2000 46N50, 47Dxx, 47F05.     

Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

By means of a noncommutative differential calculus on function space of discrete Abelian groups and that of the regular lattice with equal spacing as well as the discrete symplectic geometry and a kind of classical mechanical systems with separable Hamiltonian of the type H(p, q) = T(p) + V(q) on regular lattice, we introduce the discrete symplectic algorithm, i.e., the phase-space discrete counterpart of the symplectic algorithm including original symplectic schemes and the jet-symplectic schemes in terms of the discrete time jet bundle formalism, on the regular lattice. We show some numerical calculation examples and compare the results of different schemes.     

2p-Commutator on Differential Operators of Order p

We show that a space of one variable differential operators of order p admits non-trivial 2p-commutator and the number 2p here can not be improved.